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EDUCATED, INTELLIGENT MEN, 



WHO ARE INTERESTED IW THIS SUBJECT, BUT NOT 
THOROUGHLY ACQUAINTED WITH THE THEORY 
UPON WHICH LIFE INSURANCE CALCULA- 
TIONS ARE BASED, AND THE PRINCI- 
PLES UPON ^WHICH THE BUSI- 
NESS IS FOUNDED. 



BY aUSTAVUS W. SMITH. 

ATLANTA, GEORGIA, 1869. 



NOTES ON LIFE INSURANCE. 

IN TWO PARTS. 
THEORY OF LIFE INSURANCE. 

NET VALUE OF THE RISK ON" ONE DOLLAR FOR ONE YEAB.- 
WET SINGLE PREMIUM.-NET ANNUAL PREMIUM.- 
TRUST-EUND ON DEPOSIT, OR RESERVE.— NET 
COST OF INSURANCE.-VALUATION OF 
POLICIES. — FORMULAS AND TA- 
BLES USED IN MAKING NET 
CALCULATIONS. 



SEC03SriD 



PRACTICAL LIFE INSURANCE. 

LOADING.-EXPENSES.-SURPLUS.-ADDITIONS TO POLICIES. 
LOANS IN PART PAYMENT OF PREMIUMS, 
AND GENERAL COMMENTS. 



" The rate of PREMIUM which MUST BE charged, in order to carry out an Insurance 
contract, is the problem which stands at the threshold of Life Assurance.*' —Dr. FARR. 



BY aUSTAVUS Wl SMITH, 

ATLANTA, GJEOMGIA, 1S69. 



FRANKFORT, KY.: 

PRINTED AT THE KENTUCKY YEOMAN OFFICE. 

S. I. M. MAJOR. 

1870. 



ilt^qV 



/*. 



ACi 



Entered according to Act of Congress, in the year 1870, 

BY GUSTAVUS W. SMITH, 

In the Office of the Librarian of Congress, at Washington. 



**Ib tise absence of a popular treatise in whicia tbe science of Life Insurance 
Is faithfully and thoroughly interpreted, these annual attempts to throw some 
light, both on the theory and practice, may be of service." — Elizur Wright^ 1865. 

"The wonderful growth of Life Insurance in this country, since it has been 
explained and popularized by such essays as those of Professor Wright, seems 
to prove that, like all good things, it prospers in Light rather than in Darkness." 

D. Parks Faekler, 1868. 

"Does the system itself rest on principles and laws so certain and stable as to 
justify a reasonable conviction that, if the system is fairly and honestly admin- 
istered, the bread that is cast on its waters will be surely found, though after 
m&ny days ? " — John, E. Sanford^ 1868, 



INTRODUCTION. 



The following notes are not addressed to Officers, Direct- 
ors, or Agents of Life Insurance Companies; because, in 
the absence of positive proof to the contrary, it is reasona- 
ble to assume that these persons understand the theory and 
principles of the business they have undertaken to control. 
In case some of them are not thoroughly informed already,, 
they can no doubt readily procure information from expe- 
rienced, trained "Experts," Life Insurance Actuaries. 

It is far from my purpose, how^ever, to intimate a desire 
that Life Insurance Officers and Agents shall not read and 
study these "Notes" if they like. I only disclaim any inten- 
tion to attempt to guide or teach those now in the business. 
If an Officer, or Agent of a Life Insurance Company, feels 
there is something connected with the business he does not 
clearly understand, and it is inconvenient for the Actuary, o? 
consulting Actuary of his Company, to explain it to him,, 
he may, perhaps, find some of the principles for which he is 
searching, explained in the following Notes. 

It has been well said : " All men should study their own 
profession, not only with a view to its own peculiar interests,. 
but also as a part of the general mechanism of the world. '^ 
In other words^ it is essential that everp man shauld understand 
the business in which he is en^as;ed. Life Insurance forms no- 
proper exception to this general rule. 

In the summer of 1869, I was requested to become a State 
Agent for a Life Insurance Company. I determined not to 
accept the oifer, until I could, at least, satisfy myself that 
there was no " mystery in the art " so deeply hidden in " ab-- 
struse science" that I could not comprehend it. This led me 
to an examination of the published official reports of the 
Insurance Commissioner of Massachusetts, and of the Super- 
intendent of the Insurance Department of the State of New 
York. The first subject to which my attention was directed^ 



Introduction^ 5 

was an essay on the "Contribution Plan of apportioning 
dividends." The "algebra" of this business was found to 
be of the simplest elementary character; but the underlying 
meaning and sense of the thing, was not so clear at first 
sight. Although the "algebra" and the reasoning upon the 
"reserve," "net annual premium," and "cost of insurance," 
were mere «, 6, c, and followed as a matter of course, pro- 
vided the nature and amount of the quantities used were 
understood, I am bound to say that I knew nothing what- 
ever of the "Contribution Plan" after reading the essay; 
simply because I knew nothing of the quantities referred 
to therein, viz: the "reserve," the "net annual premium," 
and the "cost of insurance," This led me to commence 
further back.- I finally started at the beginning, viz : " the 
amount that will, if placed at a given rate of interest, pro- 
^duce one dollar certain at the end of the year, when the 
interest for one year is added to the principal," and then 
endeavored to follow, step by step, to. the end. With the 
guides I had, and the "lights" before me, I experienced 
some difficulty in keeping upon the right road; and when, 
at times, I wandered from it, there was trouble in finding it 
again. 

If by writing and publishing these " Notes on Life Insur- 
ance," I may be instrumental in bringing this important 
subject more clearly before the intelligent, educated busi- 
ness men of the country, thus enlarging the number of the 
initiated, and bringing new minds into the discussion, I will 
be more than compensated for the trouble with which I 
met in trying to follow the "lights" and "signs" set up 
alongnhe road 1 traveled when searching out the " mysteries 
of the art," 

I have persistently endeavored to convince the general 
reader that the " reserve " of which we hear so much, is not 
" CASH CAPITAL j" uor is a large " reserve " any evidence of 
superabundant riches, or strength, on the part of the Life 
Insurance Company; in excess of the means, necessary to 
enable the Company to comply with its obligations, and pay 
its policies at maturity. The fund generally designated 
by Life Insurance writers as " reserve," is an accrued 
liability— a debt. If a Life Insurance Company desires to 



B IntroducUon. 

give a clear and candid exposition of its affairs, it will 
accurately compute this " reserve," acknowledge it to be 
a debt, make it prominent in the statement of its liabili- 
ties, and then show, if it can, that the Company has bona 
fide assets to meet all its liabilities. 

Without further preamble, the general reader is respect- 
fully referred to the following pages, in which, it is believedy 
he will find the "key" that will enable him, without great 
difficulty, to investigate and understand the principles upon 
which the theory of Life Insurance is founded, and judge 
correctly of the practical working of this business. The 
arithmetical examples are introduced solely for illustra- 
tion, and the results are not to be taken as bald, isolated 
facts. If errors have been made in the calculations, they 
can be easily detected and corrected by a good computer y 
because the principles and the data upon which all the 
computations are made are fully explained before the arith- 
metical illustrations are given. 



NOTES ON LIFE INSURANCE 



Life Insurance will bear the closest scrutiny. It needs it. 
There is no magic in the art, and no mystery ought to exist 
in reference to a subject of such magnitude and serious 
importance. There are now more than $2,000,000,000 in- 
sured upon lives in this country ; and the welfare of more 
than 600,000 families is largely dependent, in case of the 
death of their natural protectors, upon the prompt payment 
of the amounts insured. 

It would seem that the enormous sum of money involved, 
as well as the nature of the obligations incurred by these 
companies, ought to attract the attention of intelligent, bus- 
iness men ; not for the purpose of making money ; not 
necessarily for the purpose of taking out policies upon their 
own lives ; but in order to comprehend the nature of this, 
comparatively new, and already gigantic, element in busi- 
ness — recently introduced by modern civilization — and to 
judge correctly of the various interests that are directly 
and indirectly affected thereby. 

Life Insurance is rapidly increasing, and must produce 
either great good, or great evil. It is essential that its pecu- 
liarities be clearly understood by those directly concerned; 
and all intelligent men in the country will readily under- 
stand that $2,000,000,000 in any one business, is a sum, 
which, once jeopardized, might injuriously affect all other 
values. 

A discussion of the general principles of Life Insurance, 
and the relative merits of the various systems upon which 
it is conducted, is certainly admissible ; and it is hoped the 
following Notes may be of some interest to those who are 
not already conversant with the subject. 



ABBREVIATIONS 



Mathematical notation will not be of material as&istance 
to the general reader in forming definite and accurate ideas 
of the principles and quantities that are used in Life Insur- 
ance calculations. But, after correct ideas on these points 
are once clearly formed in the mind of the reader, it will be 
convenient, for both writer and reader, to represent certain 
quantities by symbols, instead of constantly repeating the 
whole of the words needed, in the first place, to give a 
definite idea of the nature and the manner of determining 
the quantity in question. 

In the following pages, for instance, the method of cal- 
culating the amount of money that will, if paid in hand, 
at any named age, be just sufficient to insure one dollar 
for whole life, will be dwelt on at considerable length. 
Afterward, this amount will be called " net single pre- 
mium," and will, for any age a-, be represented by the 
symbol sP^., and the general reader will, by this symbol, 
be enabled to recall a definite idea of the process for 
obtaining this net single premium, and keep in mind an 
exact idea of what this premium is, as well by seeing the 
symbol ^P^, as by having the whole process again fully 
elaborated by the repetition of half a dozen paragraphs. 

The amount that will, if paid annually, at the beginning 
of the year, insure one dollar to be paid to the heirs of 
the insured at the end of the year in which he may die, 
is also dwelt upon at length. It is called the " net annual 
PREMIUM," and for any age x, is represented by aV^. 

The money value in hand, at any age x, of a future life 
series of annual payments of one dollar each, and the method 
used for obtaining this value, is fully explained hereafter. 
The first payment, of one dollar, is to be made in hand, and 
one at the commencement of each following year : provided^ 
the person is alive to make the payment ; and the present 



Ahhreviations. 9 

value at age :r, of a future whole-life series of annual pre- 
miums of one dollar each, is represented by -A^. 

There is, in Life Insurance, a peculiar and very impor- 
tant element, which is, by writers on this subject, called 
by various names. It is often styled the "reserve," or 
*' reserve for reinsurance." Sometimes it is called "net 
premium reserve," at others " net value," or " true value." 
It is really an amount of money belonging to the policy- 
holder, and is held by the Company in trust. This " trust 
FUJTD deposit" is, in the following Notes, represented by the 
letters (T. F. D.), and the amount that should be in deposit 
at the end of n years from the date of a policy, taken out 
at the age x, is represented by (T. F. 'D.)x^n. It will be 
seen hereafter that if a Life Insurance Company has not 
on hand the requisite amount to make up this trust fund 
on deposit, and if this amount is not placed to the credit 
of the respective policies, and regularly increased by net 
or table interest, compounded yearly, the Company will be 
unable to pay its policies at maturity. 

It should be constantly borne in mind that these 'abbrevia- 
tions mean, in every case, "a sum of money," the amount 
or value of which is readily susceptible of direct and easy, 
but sometimes tedious, arithmetical computation. 

(T. F. J).)x-\-n may, to some readers, be rather a formi- 
dable looking symbol; but when they once understand that 
it is " a sum of money ^'' intended for a very special and 
important purpose, and know exactly how to compute its 
precise arithmetical amount, and what it is used for, and 
to whom it of right belongs, and that T stands for Trust; 
P for Fund; and D for Deposit; that x stands for the age 
of the policy-holder at the time he first took out his policy, 
and n for the number of years the policy has been in force, 
it is believed that the intelligent reader will not find any- 
thing very abstruse in (T. F. D.):t-|-w, or in any other of 
the abbreviations used in these " Notes." 



PART FIRST. 



Theory of Life Insurance. 

The law of duration of human life, when applied to large 
numbers of mankind, has been very accurately determined. 
The perfect uncertainty that a. particular individual will 
survive any definite period, is equaled by the certainty that 
out of, say 100,000 persons, living at a particular age, a 
given, and well ascertained number of these, will die in each 
year, and a certain number will be living at the end of each 
year. It is the province of Life Insurance, in a pecuniary 
sense, to enable the insured to replace the uncertainty of 
his own life, by the certainty of the law governing the 
general duration of human life, and thus avert the disaster 
that might befall his family in case of his early death. 

The insured, upon paying an annual premium which will 
cover the risk upon his individual life, enables the Company 
to pay, with certainty, to his heirs, a stipulated sum when- 
ever he may die. It is proposed to discuss the manner in 
which this is done by Life Insurance Companies, 

Life Insurance calculations are, as a general rule, made 
on the supposition that the anlount insured is one dollar; 
and having obtained the premium that will insure one 
dollar, it is an easy matter to determine the premium for a 
similar kind of policy for any other sum. 

In order to determine what amount will be sufficient to 
insure one dollar, to be paid to the heirs of a person at the 
end of one year from the date of the payment of the pre- 
mium — provided the insured dies during the year — we must 
have at hand a Table of Mortality, and a safe rate of 
interest must be fixed upon. 

The Table of Mortality used in these Notes is the Actu- 
aries', and the rate of interest assumed is four per cent.; 
but, for the present, we may leave the rate of interest un- 
decided and call it r. We will thereby deduce general 



Life Insurance. 11 

rules for the calculations, and can afterwards assign to r 
any particular value that may be decided upon. The first 
question to be settled is : what is the amount of money that 
will, if invested at a rate of interest r, produce one dollar 
certain at the end of one year? That is to say, the principal 
and the interest together must amount to one dollar at the 
end of one year. 

The Amount that will Produce One Dollar in One Year. 

Suppose we represent this amount by v. Then r times v^ 
divided by 100, will represent the interest on -y, at the rate 

T V 

r for one year— that is, — -is the interest. Add this to the 

principal, which is v, and the two together must, from the 
condition imposed, be equal to one dollar. Therefore, we 

T V 

have the equation v-\- =$1. Multiply both members of 

this equation by 100, in order to clear it of fractions, 
and we have lOOt^-j-^^^lO^^ or, i;(100-|-r)=100 ; hence 

qj=: — . We see from this last equation, that the 

100+r ^ J 

amount of money v, which will, at any rate of interest r, 
produce one dollar in one year, is obtained by dividing 100 
by 100 plus the rate of interest. Give to r any assigned 
value; substitute for r in the above equation the value thus 
assigned, and the problem becomes one of very simple arith- 
metic. For instance, suppose the rate of interest is four 
per cent., then r is equal to four, and the second member 

of the equation becomes ^ — , which is the same thing as 

By performing the division indicated, we have v = 

$0.961538; and this is the amount of money which will, 
at four per cent., produce one dolljlr in one year. By 
giving to r any other assigned value, we can, in a man- 
ner entirely similar, find the amount which will, at this 
newly assigned rate of interest, produce one dollar at the 
end of one year. 



1$ J^otes on 

The Amount that will Insure One Dollar for One Year. 

But let us now suppose that, instead of having to pay 
the one dollar, certain, at the end of one year, it had been 
agreed that the person, or company, was to pay the dollar 
at the end of one year only on condition that the insured 
died during the year. 

If it can be determined accurately what chance, or prob- 
ability, there is that the insured may die during the year, 
we have only to m.ultiply the present value of the one 
dollar, to be paid certain at the end of one year, by the 
fraction which expresses this chance, or probability, in 
order to determine what it is now worth to insure oncrs 
dollar, to be paid to the heirs of the insured, in case he 
dies within one year from the date of the transaction. 

The question, whether a person now in good health may 
die within a year, if applied to a single individual, is to the 
last degree uncertain ; but close observation of accurate 
statistics, .has established the fact, that there is a fixed gen- 
eral law governing the duration of* human life, when ap- 
plied to large numbers of mankind ; and it is known, that, 
of any large number of people living at any particular 
age, a certain proportion of these will die each year. 
For instance, suppose we take 100.000 persons living at 
ten years of age ; during the year between age ten and 
age eleven, 676 of these persons will die, and there will 
be 99,324 of them living at the age eleven. In the year, 
between age eleven and age twelve, a given number will 
die, and in each successive year a given number will die, 
until the last passes away in death. 

The results of statistical observation upon the duration 
of human life have been carefully tabulated and combined 
in what are called " Mortality Tables." In these tables are 
recorded the number living at each age, and the number of 
these that will die within the following year. Now, suppose 
that there are a large number of persons of any given age, 
insured in a particular Company, enough to cause the aver- 
age mortality amongst the insured to conform to the law 
of mortality, as expressed in the table. We can, by using 
the Mortality Table, not only find the chance or probability 



Life Insurance. IS 

that a particular individual of those thus insured may die 
during any year, but in applying this to all the insured, the 
business would be entirely divested of that element of pure 
gambling upon chances, upon w^hich it M^ould be founded, 
if applied to a single individual, or even a small number 
of individuals. There is nothing more true than that " Life 
Insurance seeks breadth of basis, and can not be safely 
cooped within narrow limits." 

With an accurate Mortality Table at hand, we could find 
the number of persons living at the age of the person who 
desires to have, say one dollar, insured to be paid to his 
heirs at the end of one year, in case he dies during the 
year. Of the number living at the beginning of the year, 
the table gives the number that will die during the year. 
Now, on the supposition that, of the whole number living 
at the beginning of the year, each has as good a chance 
of living, or dying, as another, we obtain the chance or 
probability that any given one of the number will die 
during the year by dividing the whole number of deaths 
during the year by the whole number living at the begin- 
ning of the year. This will give the fraction which repre- 
sents the chance or probability that the insured may die 
during the year. 

The Fraction that Represents the Chance or Probability 

THAT THE InSURED MAY DiE DuRING ANY YeAR. 

In illustration of the calculation of the chance or proba- 
bility that a person may die during any given time, let us 
suppose that out of one hundred persons condemned to be 
shot on a given day, all are reprieved, for the day, except 
one, and the one to be shot is to be the one who draws the 
black ball out of a box containing ninety-nine white balls, 
and one black one. The chance, before the drawing, of any 
particular man's getting the black ball, and, therefore, his 
chance of being shot that day, is one only out of one hun- 
dred, or -— - of a certainty. If two men had to die that day, 

each individual's chance, before the drawing, of getting a 
black ball, would be twice as great as it was before, for now 



IJp J\^otes on 

there are two black balls and only ninety-eight white ones 

2 

in the box. The chance, in this case, would be of a 

*' ' 100 

Q 

certainty ; for three persons , and so on to the limit of 

one hundred one hundredths, which would make it certain 
that each man would be shot, because all the one hundred 
balls were black. 

To apply this principle to Life Insurance, and to show 
"that the amount that will insure one dollar to be paid 
certain at the end of any year, when multiplied by the 
fraction which represents the chance or probability that the 
person will die during the year, gives what it is worth to 
insure one dollar, to be paid at the end of the year, in case 
the insured dies during the year." Let us suppose that out 
of one hundred persons alive at the beginning of the year, 
it is known that one of them, and one only, will die during 
the year. The value of one dollar, to be paid certain at the 
end of one year, is represented by v, which, in the particular 
case of interest at four per cent., has been shown to be 
$0.961538. The chance that the person will die during the 
year, is one out of one hundred, or the one one-hundredth 
part of a certainty. To make it certain that his heirs will 
obtain one dollar at the end of the year, he must advance 
$0.961538 at the beginning of the year. But the one dollar 
is not to be paid certain ; it is to be paid only in case he 
dies. Suppose the whole one hundred persons are insured; 
then, since there is but one to die, and but one dollar to be 
paid, each person will have to give only the one-hundredth 
part of V in order to make up the whole of v. Therefore, 

— — , multiplied by v, is what each man would have to 

pay. In case it is known that two persons out of the one 
hundred will die, the amount requisite to effect the insur- 
ance will be twice as much as before, because two dollars 
must be paid certain at the end of the year. The chance 
or probability that each person maj^ die during the year is 
twice as great as in the first case, and is therefore two 
out of one hundred; and the amount that each person will 

2 

have to pay for his insurance is — - multiplied by v. The 



Life Insurance. 15 

other ninety-eight persons will get nothing; they have been 
insured for one year, and have paid for that insurance. 
The aggregate of all the payments was 2v, and this, at 
four per cent., produced two dollars certain at the end of 
the year, with which to pay the heirs of the two persons 
that died. 

In case it is known that three persons of the one hundred 

will die during the year, each will have to pay of v ; and 

the chance or probability that any individual of the whole 
number will die during the year is represented by the fraction 

3 

. In case it is known that all of them will die during the 

100 • 

year, the fraction which represents the chance or probability 

that any particular individual will die, becomes , or unity. 

This represents the certainty ; and to insure one dollar to be 

paid at the end of the year, to the heirs of the insured, in 

case he dies during the year, each person will now have to 

100 ^ , . , . , , . 

pay -— - 01 Vy which is equal to v; that is to say, each must, 

in this case, pay enough to make, with interest at four per 
cent, added, the full amount for which he is insured. 

To Insure One Dollar for One Year at Age 30. 

To return to the Mortality Table : Suppose the person to 

be thirty years of age. The table shows that out of 100,000 

persons living at the age ten, there will be 86,292 living at 

the age thirty. The number of deaths between age thirty 

727 

and age thirty-one is 727. Therefore, is the fraction 

^ 86,292 

which represents the chance or probability that the insured 

will die before he is thirty- one years of age. 

The present value of one dollar, to be paid certain at the 

end of one year, has, in the case of interest at four per 

cent., been found to be equal to $0.961538. Now, multiply 

727 
this by the fraction - , — , which represents the chance or 
86,292 * • 

probability that the insured will die during the year between 



16 J{otes on 

age thirty and age thirty-one, and we have $0.00810083. 
This is the value of the risk on one dollar, or what it is worth 
to insure one dollar to be paid to the heirs of a person at 
the end of one year, in case he dies during the year, the 
age of the insured being thirty years at the time he took 
out the policy. It would require one thousand times as 
much to insure one thousand dollars as it does to insure 
one dollar, and half as much to insure half a dollar as it 
does a whole dollar. We can obtain, by using the Mortality 
Table, the fraction representing the chance or probability of 
a person dying during the year at any named age, just as 
w^e did in this case for age thirty. We have, therefore, 
already determined the means for calculating the sum that 
will insure any given amount to be paid to the heirs of the 
insured in one year, in case he dies during the year ; but 
we must know the age of the person, the rate of interest 
must be fixed, and a Mortality Table must be available for 
use in the calculations. ^ 

To Lnsure Ten Thoqsand Dollars for One Year at Age 

Forty. 

Suppose, for instance, the age of the person applying for 
insurance is forty ; the Mortality Table used is the Actuaries', 
and the rate of interest is four per cent. ; the insurance to 
be for one year; the amount of the policy ten thousand dol- 
lars, and the insured is a fair average of mankind in health, 
and prospect of longevity. What amount in hand ought to 
be paid to insure ten thousand dollars as above, leaving 
out all consideration of expenses, and having neither gain 
nor loss represented in the chances of the transaction ? 

We wdll first determine the amount that will insure one 
dollar. We have before found that the present value of 
one dollar, to be paid certain at the end of one year, inter- 
est being assumed at four per cent, per annum, is equal to 
$0.961538. From the Mortality Table we find that out of 
100,000 persons living at the age ten, there are 78,653 living 
at the age forty, and that 815 of these will die during the 
year between the age forty and the age forty-one. There- 
fore, is the fraction which represents the chance or 

78,653 ^ 



Life Insurance, 1 7 

probability that the insured will die during the year; and 
this multiplied by $0.961538, which is the present value of 
one dollar, to be paid cej^tain at the end of one year, will 
give what it is now worth to insure the one dollar, to be 
paid in case the insured dies during the year, or $0.961538 

X— ^-^ = $0.009963428. This multiplied by 10,000, makes 

7 o,oOo 

$99.63, plus a fraction of a cent, which is the net single 
premium that will insure ten thousand dollars, to be paid 
to the heirs of the person insured at the end of one year : 
provided he dies during the year ; and in case the whole 
78,653 persons had been insured, the 815. who died would 
each have had ten thousand dollars paid to their heirs, and 
there would have been nothing left at the end of the year's 
business. Every one of the whole 78,653 had been insured 
for one year in the sum of ten thousand dollars each; but 
only the heirs of those who died were respectively entitled 
to, and were paid, the ten thousand dollars. The aggre- 
gate payment of losses by death, for the year, amounted, 
therefore, to $8,150,000, and the net annual premiums $99.63 
each ; when increased by four per cent, interest, amounted 
to $8,149,996.43. The omitted decimals on a single net an- 
nual premium amounts to three dollars and some cents, 
when carried into as many as 78,653 policies. 

Present Value that will Produce One Dollar Certain in 

Two Years. 

Now the question arises, what will be the present price or 
value necessary to insure one dollar, to be paid to the heirs 
of the insured in case he dies during the second year? 

We will first find, as before, the present value of the one 
dollar to be paid certain; but in this case it is payable at the 
end of ^m;o years. We will designate the value payable 
certain at the end of two years by v\ and call the rate of 
interest r, as before. Now, if we multiply v" by the rate of 
interest ?% and divide the product by 100, we will obtain an 
expression which represents the interest on v" at the rate r 
2 



IS J^otes on 

for the first year. The interest for the first year is therefore 

TV 

represented by — -. Add this interest to the principal v" 

w" 
and we have v"A-- — , which is the sum to be placed at 

interest at the beginning of the second year. This sum v'-^- 

, multiplied by r, and the product divided by 100, will 

T I w" \ 

srive us \i^"-\- 1, which is the interest during the sec- 

^ 100\ 100/ ^ 

ond year. The original sum v\ with the interest for the 

first year and the interest for the second year added to it, is 

TV T I 

equal to one dollar: therefore we have v"A- — -i- \v"-\- 

^ ^ 100^ 100\ ^ 

)=$1. Multiplying both members of this equation by 

10,000, in order to clear it of denominators, we have 10,000* 
v"-^ 200?V+ r'v"= 10,000, or ^'(10,000+ 200r+r')= 10,000. 
10,000 
10,000+2b0r-fr^ 
which gives the value of v'\ any given or assigned value of 
r, and it becomes a question of very simple arithmetic to 
obtain the numerical value of v'\ at any assigned rate. Let 
us suppose the rate of interest is as before, viz : 4 per cent. 

10,000 



Hence, v"=^ ^ ^ — ^. Substituting in this expression, 



per annum; that is r=4, and we find, v"= 
^=$0.924556. 



10,000-f800-f 16 
10,000 



10,816 

Before proceeding further, it is well to note the fact that 
the algebraic expression for v^ in terms of r, viz : v^= 

~ — - will, when multiplied by itself, or raised to the 

second power, become v= ■ — ^\ and this is 

^ 10,000-K200r+r' 

the precise expression found above for the value of v". 

Therefore, v"=v'^', that is to say, the present value of 

one dollar, payable certain at the end of two years at 

any rate of interest ?% is equal to the present value of 

one dollar, payable certain at the end of one year, at the 



Life Insurance. 19 

same rate of interest, raised to the second power. And 
so, too, the arithmetical value of -y, multiplied by itself, 
will produce the arithmetical value of v" :, that is, $0.961538 
X$0.961538=$0.924556. 

Notice that v is always less than one dollar; its value 
is, therefore, a fraction of unity, and multiplying v by itself, 
the resulting value must be less than v. 

Present Value that will Produce One Dollar Certain in 

n Years. 

Calling the present value of one dollar, to be paid certain 
at the end of three years, v"\ and placing this at a rate 
of interest r, we find in a similar manner an algebraic 
expression for the value of v'" . Having found this value, 
an inspection of the algebraic expression will show that 
v" is equal to v raised to the third power ; the same will 
be true for the arithmetical values of v and v'" . 

In short, it is an algebraic law that the present value of 
one dollar (computed at any given rate of interest), to be 
paid certain at the end of one year, will, when raised to a 
power, the exponent of which is n, be equal to the present 
value of one dollar, to be paid certain at the end of n 
years, interest being compounded annually. 

ISTET SX2^TO■IL.E lE^ IR E 3S^ I TJ 3VE - 

To Calculate the Net Single Premium at Age a:, that will 
Insure One Dollar for Whole Life. 

We are now ready to calculate the net single premium 
that will insure one dollar, to be paid to the heirs of a 
person at the close of the year in which he may die ; that 
is, to determine what sum paid in hand will, on the suppo- 
sition that the Mortality Table is correct, and the assumed 
rate of interest is always realized and compounded yearly, 
be the exact equivalent of one dollar insured, to be paid 
at the end of the year in which the person insured may 
die. 

We have seen how to calculate the present value of one 
dollar, to be paid certain at the end of one, two, three, or 
any number n of years. We have seen, too, that the Mor- 



20 Jfotes on 

tality Table furnishes the data for determining' the fraction 
which expresses the chance or probability of the insured 
dying in any year. Let us suppose, now, that the person 
desiring to be insured is thirty years of age. We have 
previously seen that the value of v is equal to $0.961538; 
and that the fraction which expresses the chance or prob- 
ability of the person's dying between the age thirty and 

727 
the age thirty-one, is : and that the value of one 

^ ^ 86,292 

dollar, to be paid certain at the end of one year, multiplied 
by the probability of the insured dying during the year, 
gives for the amount that would insure one dollar for one 
year, $0.00810083. * 

Now, calculate in a similar manner the present value that 
will insure one dollar, to be paid at the end of two years, 
provided the insured dies during the second year. In this 
case V becomes v^. The table gives the number of deaths 
between age thirty-one and age thirty-two ; this number, 
divided by the whole number living at age thirty, will give 
the present chance or probability that the insured will die 
during the second year. The calculation is made in a 
similar manner for the third year, fourth year, and for 
every year up to and including age ninety-nine, which is, 
by the Table of Mortality we are using, the practical limit 
of human life. All these respective yearly present values 
are added together, and the sum gives the amount of the 
net single premium that would just insure one dollar, to be 
paid to the heirs of a person at the end of the year in 
which he may die ; the age of the person at the time of 
taking out his policy being, in this case, thirty years. 

For any age different from that assumed, the numbers 
taken from the table will be found opposite to the age, and 
the fraction representing the chance or probability of the 
person's dying varies with the age. But v is always the 
same, for the rate of interest assumed, four per cent., viz : 
$0.961538 for the first year, and the square of this for the 
second year; and the present value of one dollar, to be paid 
certain at the end of n years, is equal to f , raised to a power 
whose exponent is n. 

Having found the net single premium that will insure one 
dollar for life, any other amount at the same age will be 



Life insurance, Bl 

insured by a proportional net single premium. We have, 
therefore, shown how the net single premium may be calcu- 
lated that will insure at any age any given amount, to be 
paid to the heirs of the insured at the end of the year in 
which he may die. It requires a good deal of plain arith- 
metic. 

In case the insured is not a very old man at the time of 
taking out his policy, a great deal of ciphering will be nec- 
essary in order to make the calculation for determining the 
net single premium that will insure one dollar for whole life. 
For instance, at age twenty the calculation mast be made for 
each year separately, from twenty to ninety-nine, inclusive; 
and these yearly values must all be added together. At age 
ninety-nine, however, in case a person at that age desired to 
be insured, the calculation for the net single premium is 
readily made, and with very little actual ciphering, because 
ninety-nine is the limit of the duration of human life, accord- 
ing to the Table of Mortality we are now using. We will, 
therefore, have to calculate for but one year, and the fraction 
which represents the chance or probability that the insured 
will die during the year, is equal to one divided by one, or 
unity. 

It follows, therefore, that the amount required at age 
ninety-nine, to insure one dollar to be paid to the heirs of 
the insured at the end of the year in which he may die, is 
equal to the amount in hand that will, at the end of one 
year, when table or net interest is added, be just equal to 
one dollar ; that is to say, at age ninety-nine, the net single 
premium that will insure one dollar for whole life is equal to 
V. From this it is seen, that, in case only the value of the 
actual risk on one dollar for each successive year is paid, 
the price for each year increases, until at age ninety-nine, 
the amount the insured will have to pay at that age is the 
amount that will, at table interest, produce one dollar certain 
in one year. 

Insuring for each Separate Year, and Insuring by Net 
Single Premium for Whole Life in Advance. 

It is therefore seen, that, in the later years of a long 
life, if the method of insuring for only one year at a time 



f^ J^Totes on 

is followed, the yearly premium will finally become almost 
equal to the amount of the policy. This plan of paying 
each year only the value of the risk during that year on 
one dollar will, in time, become very burdensome to the 
policy-holder, and the result will be very unsatisfactory to 
those who live for a long period. 

On the other hand, the payment of a net single premium 
for whole life insurance requires a very large sum in ad- 
vance. By this single payment the insured places at once 
in the hands of the Insurance Company an amount which 
is sufficient, when increased by net interest for the first 
policy-year, to pay the actual value of the risk on his policy 
during that year, and leave in the hands of the Com- 
pany at the end of the first year an amount equal to the 
net single premium due to the age x-\-\', and so for each 
successive year to the limit of the Table of Mortality. 

To obviate the difficulties inherent in these two methods 
of insurance, viz : in one case a constantly increasing pre- 
mium, in case the insured pays each year only the value 
of the risk during that year ; in the other a very heavy 
payment made at once in advance, that covers the value 
of the risk during every year; a system of equal annual 
payments has been devised, so adjusted as to be the precise 
equivalent in money value of the net single premium : 
this, on the supposition that the Mortality Table is exact, 
and that net or table interest is always realized and com- 
pounded yearly. By this method of equal annual payments 
the insured, during the earlier years of his policy, pays 
more than the value of the risk each year. This excess of 
payment is increased by table interest, compounded, and 
in the later years of a long life will be just sufficient to 
counterbalance the deficiency^ during these years, in the 
annual premium, 

Having seen how to calculate the net single premium 
that w^ll insure any given amount to be paid to the heirs 
of a person at the end of the year in which he may die, 
we will now proceed to show what the value of each one 
of a series of equal annual premiums must be in order to 



Life Insurance. "W; 

insure one dollar to be paid to the heirs of a person at 
the end of the year in which he may die, the first payment 
being made in advance, and one at the beginning of each 
succeeding year ; provided the person is alive to make 
the payment. 

Value of a Life Series of Annual Premiums of One Dol- 
lar Each. . 

To enable us to do this, we will first find the present value 
of a life series of annual payments or premiums of one 
dollar, the first in advance. The first term of this series 
is unity, or one dollar, because the first payment is in ad- 
vance. The second term is equal to the present value of 
one dollar, to be paid certain at the end of one year, mul- 
tiplied by the fraction which represents the present chance 
or probability of the person being alive at the end of the 
year to make the payment. We will suppose that the 
person is thirty years of age. In the table we find the 
number living at age thirty is 86,292 ; and the number 
living at thirty-one is 85,565. We therefore obtain the frac- 
tion representing the chance or probability of the person 
being alive to make the second payment, by dividing the 
number living at thirty-one by the number living at thirty, 

85 565 

which is — ^ . This multiplied by v. which, as before 

86,292 r J J 

found, is equal to $0.961538, will give the present value 

of the second payment; it is $0.953437. 

To obtain the present value of the third payment, we have 
the present value of one dollar, to be paid certain at the end 
of two years, equal to i;^. From the Mortality Table we find 
the whole number living at the end of two years, and by 
dividing this by the whole number living at age thirty, we 
obtain the fraction which represents the present chance or 
probability that the person will be alive to make the third 
payment. 

Then multiply v"^ by this fraction, and we have the present 
value of the third payment. In a similar manner the calcu- 
lations are made for each year up to, and including, ninety- 
nine years of age. All these respective yearly present 



^^ ?fot6S on 

values are added together, and their sum gives us the present 
value of a life series of annual payments of one dollar, the 
first in advance — the age thirty years. By a process entirely 
similar, the calculation can be made for any other age. We 
have, therefore, shown how to calculate the value at any 
age of a series of annual payments of one dollar — the first 
payment being made in advance, the others to be paid at the 
beginning of each succeeding year; provided the person is 
alive to make the payments. It is clear that the present 
value just found bears the same proportion to an annual 
premium of one dollar as any other present value or amount 
in hand must bear to its equivalent annual premium. In 
other words, if, at any age, it is found that the present value 
of a life series of annual premiums or payments, each equal 
to one dollar, is a given sum, then the present value of a 
similar series of payments or annual premiums, each equal 
to two dollars, will be twice as much; and the present value 
of a similar series of annual premiums, of half a dollar 
each, will be half as much, and in proportion for any other 
similar series of annual premiums. 

If we once find by arithmetical calculation, as above, the 
present value of a life series of future premiums of one dol- 
lar each, at any given age, we have only to multiply this 
value by the amount of any other given annual premium in 
order to find the present value at that age of a life series of 
these premiums. 

We represent the present value of a life series of future 
annual premiums or payments of one dollar, at any age, by 
A, and A^ represents this value at the age x. Call the 
net single premium that will insure one dollar for life ^P, and 
let sV^ represent the net single premium that will insure one 
dollar for life at the age x. Since an amount A^ in hand is 
the equivalent at the age :r of a future life series of annual 
premiums of one dollar, an amount ^P^ in hand will be the 
equivalent present value at same age of a proportional similar 
series of future annual premiums. In other w^ords, A^ is to 
sV^ as one dollar, the equivalent annual premium of A^ is 
to the equivalent annual premium of ^P^.. This proportion 

^P 

gives us — ^ as the general expression for the value of 



Life Insurance. 25 

an annual premium ; the equivalent in hand for the future 
series of which is .^P^.. 

But ^P,, in hand will insure one dollar for life at the age 
X ; therefore a life series of annual premiums, each equal to 

sV 

I*', will insure one dollar for life at age x. We represent 

A, 

this net annual premium by aV ; and a?^ represents the net 

annual premium at the age x ; and we have «P^=— ^. That 

is to say, the net annual premium, «P^, that will insure one 
dollar, for life at the age x, is equal to the net single pre- 
mium that will insure one dollar for life at the same age, 
which is 5P^ divided by the value at that age of a future 
life series of annual premiums of one dollar, which we have 
agreed to call A^. 

Relation Existing Between ^P^ and A^. 

It is well to show here the peculiar relation existing be- 
tween the value at any age of a future life series of annual 
premiums of one dollar, and the net single premium that will 
insure one dollar for life at the same age. This net single 
premium is obtained by considering each year separately, 
and first finding the present value or amount that will^ at 
net interest, produce one dollar certain at the end of each 
year, and then multiply this value by the fraction which 
represents the present chance or probability that the insured 
may die in each year, and adding these results together. 

Let / represent the number living, then 4 will represent 
the number living at the age x, and 4-|-l will represent 
the number living at the age x-\-\, &c,, (fee. The number 
of deaths in any year is equal to the number living at the 
beginning of the year, minus the number living at the end 
of the year. The number of deaths during the year, be- 
tween the age x and the age x^\^ will then be represented 
by 4"~4+i 5 ^wd for the year between the age x-\-\ and 
the age a: -1-2, the number of deaths will be represented by 
4+1 — 4+2- With this notation explained, w^e can, by re- 
ferring to the method previously indicated by which to 
obtain the net single premium that will insure one dollar, 



^6 Jfotes on 

to be paid to the heirs of a person aged x, at the end of 
the year in which he may die, at once write out the fol- 
lowing equation : 

^P ^'v^ 44-1 ) I ^ (^^-l 4-1- 2) I ^'(4-1-2 L+ d)_i ^^ 

This series of terms is continued up to age ninety-nine. 
By separating the positive and negative terms, and placing 
all the positive terms of the numerators together, and divid- 
ing them by their common denominator, and arranging the 
negative terms in a similar manner, the above equation 
becomes, by placing i; as a common factor of all the posi- 
tive terms : 
p 4 +^4 4-1 -|-*^ "4 +2 4"? &c., carried out to age ninety-nine.. 

't'4^l4-'^'^4-^2^~^^4-j-3^~5 &c., carried out to age ninety-nine. 

" C 

Now, by using the same notation, and bearing in mind 
the manner in which the present value of a life series of 
annual payments of one dollar, the first in advance, was 
calculated, we are enabled to w^ite (noticing that tJie first 
'payment of one dollar is expressed in the first term of the ex- 
pression by -, which is equal to unity, and can, therefore, be 

written to represent the one dollar paid in advance), the following 
equation ; 
, 44-^44-1-4-^^4 '^-1-) &c., carried out to age ninety-nine. 

^'-^ \ \ 

By comparing this value of A^ with the equation that 
gives the value of sV ^ just above, it is seen that the positive 
term in the expression for that value is equal to t'A^., and 
that the negative term would exactly correspond with the 

value of A^. if unity or —were added to this negative term. 

Therefore, if the negative term, with unity added to it, is 
equal to A,., the negative term as it stands is equal to A^ — 1. 
By substituting 'in the equation that gives the value of 
sV ^, the expression A^ for its equivalent terms, we have — 

The Formula Used in Calculating the Net Single Premium. 

^P,=t;A,-(A-l)=l + (i;-l)A,= l-(l-^)A,. 

sY 
But we have previously seen thataP^=-— ^. Substituting 

A- 



Life Insurance. 27 

in this equation, in place of ^P^, its value just obtained in 
terms of v and A, and we have — 

The Formula Used in Calculating the Net Annual Premium. 

aV== 5^— — - — -=-r— — (1 — '^•) This is the formula used 

A., A. ^ ^ 

in calculating the net annual premium that will insure 

one dollar for life at the age x. If we can obtain the 

value of A at any age, the net single premium and the net 

annual premium to insure one dollar for life at that age 

may be readily obtained from the above equations, which 

give their respective values in terms of A and v. 

The Numerical Value of A^. 

Referring to the equation that expresses the value of A^,, 
and multiplying both the numerator and denominator of the 
fraction, which expresses that value by v"^ — which will not 
change the value of the fraction — we have a new equation 
for A^, as follows : 

. ?;%-|-'?^''"^^4+i-|-^'''^"4-f2-1~5 &^c., to, and including, age 99. 

Let us now suppose that the age is ninety-nine years, 
which is the limit of the duration of human life according 
to the Table of Mortality we are using. The equation 

would become in this case A^=-^or Agy^— ^^=1 ; because 



"99 



this is the last as well as first term of the series ; and, 
placing this value of A^ in the expression for «P^, which is 

^— — (1 — v), the net annual premium that would insure one 

A.c 

dollar to be paid to the heirs of a person aged x at the end 

of the year in which he may die, it becomes aP,9= (1 — v) 

= 1 — (1 — v)^=v\ that is to say, the net annual premium that 
will insure one dollar to be paid to the heirs of a person 
aged ninety-nine, at the end of the year in which he may 
die, is equal to the present value of one dollar, to be paid 
certain at the end of one year, interest being assumed at the 
rate r, which, in our calculations, is four per cent. This 



^8 Jfotes on 

ought to be so under the assumptions made, because the 
person, according to the table we are using, must die cer- 
tain during the ninety-ninth year. 

When x=Q9, the equation becomes thenA.j9=— ^. Sup- 
pose x=98, A,s= — '^ ^^ Suppose a:=97, we have A9^= 

— T^- -} and thus diminishing the age each year by 

one, it is seen that the numerators not only have for their 

first term the denominator pertaining to that year, but the 

second term of the numerator is the denominator of an 

age one year greater; and the third term of the numerator 

is the denominator of an age two years greater; and so 

for other ages. It follows from this reasoning, that if we 

call the denominator at any age x, D^., that the numerator 

will be expressed by D^4-D,._^i-+-D^^2-l-I^a-^?,+ to 

Dyg. If vve call the sum of all these terms of the numer- 

N 
ator at the age x, K,., we will have the equation .A,;=-^. 

These numerators 'N^, and denominators, D^, have been 
accurately calculated for the different ages, and the results 
coi-rectly tabulated. The table gives opposite the respec- 
tive ages, the numerators in a column headed N^, and the 
denominators in a column headed D,,. Then, in order to 
obtain, by using these tables, the present value, at any age, 
of a life series of annual payments of one dollar, the first 
in advance (having assumed the Actuaries' Table of Mor- 
tality and four per cent, interest per annum, compounded 
yearly); we look in the table opposite the given age, and 
in the coluinn headed N_^. we find the numerator, and in 
the column headed D^ we find the denominator ; and we 
thus obtain the fractional expression which is the value 
sought for. For example, let the age be taken at thirty 

years: the equation A^=~, becomes for this case A3o=I^— . 

In the table opposite age thirty we find the numerator in the 
N_, column ; it is 479,951.6; and in the D^ column we find 
the denominator is 26,605.37. Therefore we have A^^i= 



Life Insurance. 29 

479 951 6 

! \ — =$18,039. In a precisely similar manner we find 

26,605.37 ^ 

the arithmetical value of A^ at any other age. 

What is the Net Annual Premium that Will Insure One 
Thousand Dollars for Life at the Age Forty-two ? 

The formula is a?^=-— — (1 — v). We must first find the 

A^ 

N 
value of K^ at the age forty-two. A^=~-j which in this 

, . N,. 231,671.7 ^^^^, ^, . , 

case becomes A4o=^= =15.621. Ihereiore the 

'^ D,2 14-,830.58 

equation is, in this case, a?^c>^ — — (1 — v). But we have 

15.o^l 

previously found v to be equal to $0.961538; therefore (1 — v) 

=$0.038462. The equation then is a?,,= — ^ — —0.038462. 
^ '-^ 15.621 

By performing the division indicated in the fraction — - 

1 t>.D/i 1 

we find this fraction to equal 0.0640164. Therefore aV^^=^ 
0.0640164— 0.038462=$0. 025554. This is the net annual 
premium that will insure one dollar to be paid to the heirs 
of a person aged forty-two years, at the end of the year 
in which he may die. Multiply this by 1,000 and we find 
the net annual premium that will insure $1,000 for same 
age is equal to $25.55. 

In a similar manner the net annual premiums for whole 
life policies are calculated at any age and for any amount. 



Trust Fund Deposit, or " Reserve." 

The "Reserve" has been well styled by the highest au- 
thority, "The Great Sheet- Anchor of Life Insurance." It 
is essential that its bearings upon the practical business 
of Life Insurance be clearly understood by all who have 
anything to do with this subject. 

We have just calculated the net annual premium that will 
insure one thousand dollars for life at age forty-two. It 
will be borne in mind that this premium is to be paid at 
the beginning of each year, provided the person is alive to 



ii 



so Jfotes on 

make the payment. At the end of the first year, or begin- 
ning of the second — supposing the insured to be alive — he 
pays the net annual premium, $25.55, and is insured for 
another year; but he is now forty-three years old. 

What is the net annual premium that will insure one 
thousand dollars for life at age forty-three ? 

Making the calculation in a manner entirely similar to 
that just gone through with for age forty-two, only sub- 
stituting age forty-three in this for age forty two in that 
example, and it will be found that a net annual premium 
of $26.58 is required to insure one thousand dollars for 
life at age forty-three. Why is it that the man who was 
insured at age forty-two, and who has been insured one 
year, and has paid for that insurance, can, at forty-three 
years of age, be insured by the Company for a less pre- 
mium than is required to insure a man of the same age, 
forty-three ; but who now takes out a policy for the first 
time in that Company. Taking for further illustration a 
still greater age, we find that at age sixty-five the net 
annual premium that will insure one thousand dollars for 
life is $74.72 ; and yet the person who took out his policy 
at age forty-two, supposing he is still alive, can be safely 
insured at age sixty-five by the Company for a net annual 
premium of $25.55, 

How is this ? Why is it that a man sixty-five years of 
age can be insured safely by a Company for a net annual 
premium of $25.55; and another man of the same age, 
probably in better health, because he has just passed a 
medical examination, can . not be safely insured by the 
Company for a less net annual premium than $74.72? It 
is not a full answer to say that one of the persons has 
been insured since he was forty-two years of age, and, 
therefore, don't have to pay so much. Nor is it very sat- 
isfactory to most intelligent business men to be told, " This 
is so, because it is thus put down in the tables." There 
are now largely over half a million of policies on lives in 
force in the United States, insuring more than two thou- 
sand millions of dollars ; and it would seem that the 
amount involved, the nature and character of the obliga- 
tion incurred, and the immense interests that are directly 



Life Insuj^ance. 31 

and indirectly involved in Life Insurance business, as it 
stands in this country to-day, would attract the close atten- 
tion of intelligent men. 

However well a person may understand how to calculate 
the net annual premium, unless he also understands the 
nature and object of the "Reserve," as it is technically 
called, he knows nothing of the real business of Life Insur- 
ance ; more especially in these days, when "surplus" and 
"dividends" seem to be all the rage, both with policy- 
holders and the share-holders of nearly all the stock com- 
panies. Even the ^^ purely''^ mutual companies sometimes, 
if not generally, style their "reserve" "cash capital," when 
it is, in fact, an accrued liability of the company — a debt. 

The net annual premium is calculated to provide against 
all the probabilities and risks of the insured dying in any 
year, and of his policy becoming due ; and also the risk of 
his being alive, from year to year, to pay his annual pre- 
mium. The net annual premium, at the rate of net interest, 
assumed to be always realized and compounded yearly, is exactly 
sufficient to pay its proportion, year by year, of losses that will 
occur by the death of a certain number of policy-holders, as given 
by the Mortality Tables used ; and at the same time provide for 
the payment of the policy at the death of the policy-holder. At 
the end of each year, after the net annual premium has 
paid its proportion of the losses by death for the year, there 
must be in the hands of the Company, on account of, and to 
the credit of, each and every outstanding policy, an amount 
in money or securely invested funds, that will be in pres- 
ent value, in hand, the equivalent of an annual premium, 
equal to the difference between the net annual premium the 
insured paid on taking out his policy and the net annual 
premium he would now have to pay if he were taking out 
a new policy at his present advanced age. This amount 
that must be in the hands of the Company at the end of 
each year's business, to the credit of the respective policies, 
is variously styled, by Life Insurance writers, " reserve," "re- 
serve for reinsurance," " net premium reserve," " net value," 
and "true value." In these remarks it will be called " Trust 
Fund Deposit ;" understanding that it is an accrued liability 



32 :N'oUs on 

OR DEBT, NOT " CASH CAPITAL." Wo HOW propOSC tO SeO hoW it 

is computed. 

Referring to the value of the net annual premium already 
calculated, that will insure $1,000 for life, at age forty-two, 
which is $25 55; and the net annual premium to insure the 
same amount, at age forty-three, which, as seen above, is 
$26.58, we find the difference between these two annual 
premiums to be $1.03. 

We have previously shown that the present value of a 
life series of annual premiums of one dollar, at any age x^ 

is equal to A^. At age forty-three A^ becomes Ajt3=— -i? 

J^l^Ml.^ =15.373564. 
14,104.81 

Now, the question simply is this : If a life series of an- 
nual premiums of one dollar, at the age forty-three, is 
equal to a present value of $15.373564, what is the pres- 
ent value at same age of a life series of annual payments 
of $1.03? We find it from the proportion: 

$1 : $1.03 : : $15.373564 : the answer, 
which is equal to 15.373564X$i.03=$15.8347709. 

And this is the present value, at age forty-three, of 
a life series of annual payments of $1.03; and this $1.03 
is the difference between the net annual premium at 
age forty-three and the net annual premium at age forty- 
two; and if the Company has the $15.83 on hand in 
deposit, which is the ca-sh equivalent of this difference 
in the future net annual premiums ; this amount of cash 
in hand, together with the smaller net annual premium 
due to the age forty-two, is just the same value as the net 
annual premium due to age forty-three. This $15.83 is 
the amount that must be held on deposit in trust for the 
policy of one thousand dollars taken out at age forty-two, 
at the end of the first year of the policy; and if the Com- 
pany has it on hand, and keeps it securely invested at the 
net rate of interest, and regularly compounds the interest 
yearly, this " Trust Fund Deposit," together with the pres- " 
ent value of the future net annual premiums, will always 
keep the policy that is paying the smaller net annual pre- 
miums due to the younger age at which the holder entered 



Life Insurance. 33 

the Company, just on a par with those policies that come 
in later, or at a more advanced age of entry, and pay the 
larger annual premium due to this advanced age. 

Now, let us take up again the symbols used in Life 
Insurance, and see how the formula is deduced by which 
the amount of the Trust Fund Deposit is, in practice, actu- 
ally calculated. 

To Determine a Formula for Computing the Deposit. 
The present value of a future life series of annual pay- 
ments of one dollar is represented by A ; at any age x, it is 
A^.; and at the end of any number of years tz, from the date 
of the policy, it is represented by A^^.^. 

Now SUPPOSE A LIFE POLICY FOR ONE DOLLAR WAS TAKEN OUT 
AT THE AGE X. WhAT AMOUNT OF TRUST FUND MUST BE IN THE 
HANDS OF THE CoMPANY TO THE CREDIT OF THAT POLICY AT THE 
END OF n YEARS FROM ITS DATE ? 

The net annual premium at the age x, is aV^. The net 
annual premium at the age x-\-n, is «P^^,,. The difference 
between the two is aV^j^^ — aV^. The present value of this 
future life series of net annual premiums at age x-[-n^ is found 
by the following proportion : A future life series of net 
annual premiums of one dollar, at the age x-^n, is to its 
corresponding present value, which is A^^^, as the future 
life series of net annual premiums («?«;+„ — ^P^)) ^-t the age 
x-\-n^ is to its present value, 

or $1 : K-^j^n : : {a P^^„ — aV^ : the answer. 

It is, therefore, A^j^^{aP^_^n — «Pa)- This expression gives 
the present value, at age x^n, of a future life series of 
annual premiums («P^_|.„ — ^P^:), and this is the amount that 
must be on hand, in trust, deposited to the credit of the 
policy at the end of n years from its date, in order to make, 
at that time, when added to the value of the future «P, 
net annual premiums, an amount equivalent to the value, 
at that time, of the future al?.j^_^,^ net annual premiums due 
to the age x-{-n. 

Another Formula for the Deposit. 
Another expression may be obtained for the amount of 
Trust Funds on Deposit as follows : The net single pre- 
3 



SIf J^otes on 

mium, at any age x-^n, minus the present value, at that 
age, of the future life series of «P^ annual premiums, is the 
amount that must be on hand in deposit to cause the future 
aV^ annual premiums and the deposit to be equivalent to 
the future aV^j^^ annual premiums. From this we have the 
equation: ^P^^^ — (aP^-X Aa,^.J= the deposit at the end of n 
years from the date of the policy. Substituting for ^P^^^ its 

value, 1 — ( 1 — ?;) Aa._,.„, and for aV^ its value, -— — ( 1 — v), and we 

have 1— (1— 'u)A^^^— /— -— (1— 'y)JA^^.^=the deposit at the 

end of n years. And as the two expressions (1 — v)A.^j^n, 
have different signs, they cancel each other, and we have, 

A 

1 — „Z±!!z= "Deposit" at the end of n years. This is said 

Ax 
to be, " perhaps, the easiest working formula for obtaining 
the amount of Trust Fund that should be on deposit for 
whole life policies." 

To calculate by this formula, the "Deposit" at the end of 
the first year, for a whole life policy for $1,000 taken out at 
age 42, we have, in this case, a:=42, ?i=l, and :r-[-/^=43 ; 

. ^rA-n u , A,, -p ^ . N,3 216,841.2 

therefore, 1—— p^, becomes 1— r^. But A^g^ ''— 



A^ A,, '' D,3 14,104.81 

= 15.373564. And A,.=^i^= ' =15.621217. There- 

D42 14,830.58 

fore, 1— :^'=1— -^^5^^^^^=1— 0.984146==$0.01585. This is 
A,2 15.621217 

the "Deposit" for $1. Multiply it by 1,000, and we have 

$15.85 as the amount that must be on hand "deposited," at 

the end of the first year, to the credit of a whole life policy 

for $1,000, taken out at age 42. 

It will be noticed that this value for the " Deposit " is two 

cents more than that previously obtained; which is due to 

the fact that the diff'erence, $1.03, between the two net 

annual premiums, was not carried out to a further place of 

decimals. 

Cost of Insurance. 

The net Cost of Insurance, per year, properly chargeable 
to each policy, or, in other words, the proportion of "losses 



Life Insurance. 35 

by death," that should be paid by eacli policy out of the 
net annual premium for the year, is obtained by multiplying 
the amount the Company has at risk upon the policy during 
the year, by the fraction that expresses the chance at the 
beginning of the year that the insured will die during the 
year. This fraction is obtained by dividing the number 
of deaths given by the table, for that year, by the whole 
number living at the beginning of the year. The result 
will give the amount that the policy should contribute out 
of the net annual premium for the year, to pay losses caused 
by death that year according to the Table of Mortality. 

The " Amount at Risk " during any year, is equal to the 
amount of the policy minus the "Trust Fund Deposit" 
at the end of the year ; because the net annual premium, 
having been paid at the beginning of the year, the Com- 
pany has in its hands — of the policy-holders' money — the 
means for paying the " Cost of Insurance " during the year, 
properly chargeable to this policy ; and also for providing 
the requisite Trust Fund Deposit at the end of the year. 
Therefore, the "Amount at Risk" during any year is the 
amount insured, minus the deposit at the end of the year. 
Representing the "Trust Fund Deposit" by (T. F. D.), 
the " Deposit " at the end of n years from the date of the 
policy by (T. F. D.)^^.^, we have the Deposit for the end of 
the next succeeding year represented by (T. F. D.X.4.„^i. 
Subtracting this from one dollar, which is the amount in- 
sured, we have the " Amount at Risk " during the year 
expressed by 1 — (T. F. D.)x+n+i- Multiply this by the 
fraction which expresses the chance or probability of death 
during the year, ^nd we obtain an expression for the amount 
chargeable to the policy for " Cost of Insurance" during the 
year. This fraction is obtained by taking from the table the 
number of deaths during the year, and dividing this number 
by the number living at the beginning of the year. 

We are now considering the Mortality Table to be exact; 
the table or net interest always realized and compounded 
yearly ; that the amount insured on each policy is one dol- 
lar; that there are enough policies at any given age to 



86 J^otes on 

make the mortality amongst the insured conform to the 
general law of mortality called for by the table ; and we 
are leaving out at present all consideration of either " ex- 
penses" or "profits." We have made calculations at net 
interest, to obtain a net annual premium that will exactly 
cover all the present chances, risks, and values, year by year, 
to the table limit of the duration of human life, ninety-nine 
years of age. Therefore, when the requisite "Deposit" for 
the end of the year has been set aside for the policy, and 
the net "Cost of Insurance" during the year properly 
chargeable to this policy has been paid, there should, under 
the assumptions made, be nothing left. 

Let -^±^= number of deaths during the year, divided by 

the number living at the beginning of the year. Then 

-J^n—{T, F. D.)^4.„_^i) will represent the cost of insur- 

ance paid by this policy during the year. (T. F. D.)2._^„_|_i 
represents the Deposit at the end of the year. The above 
are the two amounts to be paid or provided for. 

At the beginning of each year there is on hand (or there 
ought to be on hand) the Deposit for the end of the pre- 
ceding year; and the net annual premium for the coming 
year is paid. Both these amounts must be increased during 
the year by net or table interest ; and out of these two 
sums thus increased, the net " Cost of Insurance " for the 
year must be paid, and the requisite Deposit for the end of 
the year must be provided. 

Let (T. F. D.)^^„(l-1-^) represent the Deposit on hand 
at the end of n years from the date of the policy, increased 
by net interest for one year. 

Let aVJ^l^r) represent the net annual premium, increased 
by net interest for one year; then, on the assumed data, we 
at once have the following equation : 



(T.F.D.U»(l+r)+aP,(l+r)- ^"(l-(T. F..D.),+„ 

(T. F. D.U.+i=0. 

This is called " the equation of equitable balance." 



+1 



Life Insurance, S7 

Let us now Calculate the " Cost of Insurance," During the 

First Year, for a Whole Life Policy of $1,000, Taken 

out at Age Forty-two. 

We have before found the "Deposit" at the end of the first 
year, for a whole life policy of one dollar, taken out at age 
forty-two, to be $0.01585. The "Amount at Risk" during 
the year will then be equal to $1— $0.01585=$0.984146. 

The table shows the number of deaths during the year is 
839, and the number living at the beginning of the year, 

OOQ 

77,012; therefore, the fraction expresses the chance 

77,012 ^ 

or probability that the insured will die during the year. 

This fraction, multiplied by the "Amount at Risk" during 

the year, gives us the *' Cost of Insurance," during the year, 

properly chargeable to this policy. 

OOQ 

Therefore, — -— X$0.984146=$0.010721=: the "Cost of 
77, Ui^ 

Insurance" on this policy of one dollar, during the year 
between age forty-two and age forty-three. Multiply this 
by 1,000, and w^e have the " Cost of Insurance " for the same 
year on a whole life policy for $1,000=$10.72. This, added 
to the am.ount requisite for the "Deposit" at the end of the 
year, which is $15.85, makes $26.57 that has to be paid or 
provided for. At the end of the preceding year there was no 
"Deposit," because the contract between the company and 
the insured had not been entered into. Therefore, for the 
year under consideration, the company has nothing but the 
net annual premium with which to pay the " Cost of Insur- 
ance" and provide the requisite " Deposit." We found the 
net annual premium to be $25.55. This is increased by net 
interest for one year. This interest has been assumed to be 
four per cent, per annum, and amounts to $1.02. This, 
added to the net annual premium, $25.55-l-$1.02=$26.57. 
The " Cost of Insurance " is paid and the " Deposit " pro- 
vided for, and there is nothing left; $26.57 — $26.57 being 
equal to zero. 

The " Reserve " Absolutely Necessary to Enable a Lifb 
Insurance Company to Pay its Policies at Maturity. 
To illustrate the manner in which the "Deposit" must 
accumulate in the earlier years of a Life Insurance Com- 



S8 J^otes on 

pany, in order to enable it to meet its obligations when 
the death claims exceed the premiums, let us suppose 
that a Company insures twenty thousand policy-holders, for 
five thousand dollars each, at age thirty. The net annual 
premium required from each person is $84.85. This, on 
20,000 policies, would make the first payment of annual 
premiums amount to $1,697,000. The net interest is as- 
sumed to be four per cent., and, for the first year, it amounts 
to $67,880. The Company has, therefore, for the first year, 
$1,764,880. By the Table of Mortality 168 of the insured 
will die during the first year ; to the heirs of each, the 
Company must pay five thousand dollars. The losses by 
death are, therefore, $840,000 ; leaving on hand with the 
Company, after all the death claims are paid, $924,880 ; 
which would be a handsome ^^ surplus ^^ at the end of the 
first year's business, but for the fact that every dollar of 
this sum belongs to the Trust Fund Deposit, and is an 
already accrued liability — a debt. 

At the end of the thirty-fourth year, the Deposit for each 
outstanding policy must be $2,464.25. The Table of Mor- 
tality shows that 11,297 of the policy-holders will be living 
at the end of the thirty-fourth year; the Company must, 
therefore, have on hand a Trust Fund Deposit amount- 
ing to $27,838,632.25. We find that 11,742 policy-holders 
were living at the beginning of the thirty-fourth year; and 
their net annual premiums amounted, in the aggregate, to 
$996,308.70. There were 445 deaths during the year, and 
the aggregate losses by death amounted to $2,225,000. 
Thus we see, in this year, the death claims exceed the an- 
nual premiums by more than one and one quarter millions 
of dollars. But the Company has on hand, in Deposit, at 
the end of the year $27,838,632.25, after having paid the 
death claims. The Company, however, is not rich, nor 
more than able to pay its liabilities, because it will surely 
take the last cent of this amount, and all the future net 
annual premiums, and compound interest regularly all the 
time, to enable it to meet and pay its now rapidly increas- 
ing death claims. 

Let us look into the accounts of the Company at the end 
of the fiftieth year. The "Deposit" on account of each 



Life Insurance. 39 

policy at the end of this year is $3,708.20; and there are 
living 3,080 policy-holders. The aggregate "Deposit" for 
the outstanding policies at this time, is $11,421,256.00. 
There were 461 deaths during the year, and the aggregate 
of policies that matured during the year amounted to 
$2,305,000. There were 3,541 policy-holders living at the 
beginning of the year, and the aggregate of the net an- 
nual premiums paid by them amounted to $300,453.85. 
We see from this that the losses by death during the year 
exceeded the net annual premiums by more than $2,000,- 
000. The "Deposit" is reduced to $11,421,256.00, which 
is less than one half the amount in " Deposit " at the end 
of the thirty-fourth year. But the Company has not lost 
money, it has only been paying its debts. At the end of 
the thirty- fourth year it had more, but it owed more. It 
had enough then, and only enough, to pay what it owed ; 
it is in the same condition now. 

At the end of the sixty-fifth year, we find the "Deposit" 
that must be in the hands of the Company to the credit of 
each policy, is $4,560.87; and there are twenty of the 
original policy-holders living. The aggregate "Deposit" 
for these twenty outstanding policies is $91,217.40. The 
$27,838,632.25 that the Company had on hand at the end 
of the thirty-fourth year is now reduced to less than $100,- 
000. But the Company has only been paying its debts to 
policy-holders — not losing money. In fact, it had none to 
lose of its own. 

At the end of the sixjy-ninth year, the " Deposit " amounts 
to $4,722.84 ; and there is one policy-holder living. He pays 
his regular net annual premium the day he is ninety-nine 
years old. The premium is $84.85. This, added to the 
"Deposit" on hand at the end of the preceding year, 
makes $4,807.69 of this policy-holder's money in the hands 
of the Company the day the policy-holder is ninety-nine 
years old. At net interest, which is four per cent., the 
interest for the year will amount to $192.31 ; and this, 
added to the amount $4,807.69, on hand at the beginning 
of the year, makes $5,000, with which to pay the policy 
of the last policy-holder in this Company. 



4<9 J^otes on 

We see that the $27,838,632.25, which the Company had 
in its possession at the end of the thirty-fourth year, be- 
longing to the policy-holders, has been paid to them. The 
policies were all paid at maturity ; the Company has noth- 
ing left. In fact, it never had a cent of its own during the 
whole time, although we have seen it the custodian, at one 
time, of nearly twenty-eight millions of dollars of other 
people's money. It owed every cent, and it paid every 
cent it owed. 

It is a marked peculiarity of Life Insurance business, 
that the annual premiums exceed the death claims for the 
first thirty or forty years ; after which time, the losses by 
death largely exceed the annual premiums. The Trust 
Fund Deposit is a fixed mathematical amount; it increases 
for each policy at the end of every succeeding year of 
the existence of the policy. And if the Life Insurance 
Company has, for each and every outstanding policy, the 
requisite " Deposit," it can pay its policies at maturity. 
This " Deposit," or " Great Sheet-Anchor of Life Insur- 
ance," is the "Sacred Fund" of the "Widows and Orphans," 
and ought to be guarded by wise and stringent laws, rigidly 
enforced by competent and honest ofiicers of the State 
Government. 

Note. — That the general reader may not sv^ppose that an extreme in the amount 
of money involved has been assumed in the above hypothetical example, used 
merely for illustrating the necessity for "accumulation" during the earlier years 
of a Life Insurance Company, in order to meet its obligations when the death 
claims exceed the premiums, the fact is here mentioned that one Company in this 
country has now over 60,000 policies outstanding, insuring over $200,000,000; 
and its Trust Funds on Deposit amount to more than $30,000,000, although the 
Company is but twenty-seven years old. 



Life Insurance. 4^ 

COMMENTS ON AMOUNT IN DEPOSIT OR RESERVE. 

Bear in mind that the "Deposit" must always be kept 
invested at the net or table rate of interest at least, and the 
interest must be regularly compounded every year, in order 
to enable the Company to pay its policies at maturity. It 
is, for this reason, not enough at the end of any year, that 
a Life Insurance Company should place to the credit of a 
policy the difference between the net annual premium due 
to the age at which the policy-holder entered the Company 
and that due to the age he has now^ attained; but the 
Company must place to the credit of the policy-holder an 
amount equal to the value in hand of a future life series 
of annual premiums, each of which is equal to the differ- 
ence above referred to. It is thus seen that in Companies 
that have a large number of policies outstanding, and pol- 
icies that have been in existence for a good many years, 
the amount is large that must be^ by the Company, placed to the 
credit of the policies, in addition to the anmial premiums paid 
by policy-holders during the year in question, in order to make 

THE POLICIES IN SUCH CoMPANY SAFE. 

The real practical meaning, therefore, of a " Rcsei^ve " 
amounting to $30,000,000 is this; The Company that has 
this amount in " Reserve " at the end of any year, has to 
pay thirty millions of dollars in addition to the annual pre- 
miums paid the following year by its policy-holders, before 
the amount is made up that will insure the payment of 
the policies at maturity. 

It is absurd to speak of a large " Reserve " as evidence 
of riches or strength on the part of Life Insurance Com- 
panies. If the Company has the requisite " Reserve," or 
Trust Fund Deposit, it can meet its liabilities; if it has not 
the requisite amount on " Deposit,^'' it is insolvent. 

Although $30,000,000 in ''Reserve'' (being the amount in 
Trust on Deposit) does not indicate riches or excess of 
strength in a Life Insurance Company beyond the mere 
ability on the part of the Company to meet its accrued 
liabilities, it makes the Company a great financial power, 
by the accumulation in the hands of its officers of this 
immense sum of money belonging to its policy-holders ; 



Jf^ J{otes on 

which amount may increase with an increase of its new 
business ; may remain permanently at this sum if the new 
business in this respect counterbalances the excess of death 
claims over premiums upon its older policies; or, if the 
Company ceases to issue new policies, the Trust Funds on 
Deposit will in time be exhausted. 

Registered Policies. 

It is a matter of great importance to policy-holders to 
have the "Deposit" guarded against all chance of accident 
or loss. To effect this, some of the Companies issue "Reg- 
istered Policies ;" in which case, the Trust Fund for the 
policy, invested in safe interest-bearing stocks, or bonds 
and mortgages, is deposited with the State Treasurer; 
and the State becomes responsible to the policy-holder for 
the safe-keeping and proper application of the funds thus 
deposited. When the real nature and importance of this 
deposit or " Reserve," as it is generally called, and the 
vast aggregate amount that it must soon attain, are well 
understood by the general public, registered policies will, 
no doubt, become more popular than they have been here- 
tofore. This would relieve the Companies, in great degree, 

FROM the care AND RESPONSIBILITY ATTENDANT UPON THE HAND- 
LING AND CONTROL OF THESE LARGE AMOUNTS OF OTHER PEOPLE's 

money. 

In case a State is willing to become responsible for the 
safe-keeping, and to guarantee the proper application of 
this fund, it would seem that policy-holders should avail 
themselves of the security thus afforded. Not that the 
State guarantees the paj^ment of the policy at maturity : 
this, I take it, a State will never do, unless it establishes 
a Government Life Insurance Company of its own. But 
some States have already agreed, and others possibly may 
hereafter agree, to become responsible for that portion of 
the fund of the Company generally known by the name 
^' Reserve ; ^'' the same which in these Notes is called Trust 
Fund Deposit. 

Whatever doubt there may be as to the proper course 
for a State to pursue in regard to this matter, there is 



Life Insurance. JfS 

hardly room for doubt that it is safer and better for the 
individual policy-holders to have their " Trust Fund De- 
posit," or " Keserve^^ guaranteed by the State, than to trust 
these vast sums solely to the officers of the Life Insurance 
Companies. 

No "Dividends" can be Made from the Net Annual Pre- 
mium AT Net Interest. 

We have seen that the whole of the " net annual pre- 
mium," at net interest, regularly compounded every year, 
is required to pay the " Cost of Insurance" during the year, 
and provide the "Deposit" at the end of the year, requisite to 
secure the payment of the policy at maturity. The " enor- 
mous dividends" made to policy-holders by Life Insurance 
Companies, and the large credits or loans so generously 
proffered, must, therefore, come from some source other 
than the " net annual premium " increased by " net inter- 
est;" and it is equally clear that the so-called "Reserves," 
accumulated by Life Insurance Companies, cannot, w\X\\. any 
propriety, be considered " cash capital." The foregoing 
remarks give an outline of the theory of whole Life Insur- 
ance. 



JfJl- J{otes on 

ALGEBRAIC SUPt^MARY OF THE THEORY OF WHOLE LIFE 

INSURANCE. 

Let V represent the amount of money that will, when 
increased by interest at the rate r, be equal to one dollar 
at the end of one year. 

Then t', raised to a power, the exponent of which is ti, 
will be the amount which will, at the same rate of interest 
r, be equal to one dollar at the end of n years — interest 
being compounded yearly. 

Let / represent a number of persons living, and l^ repre- 
sent the number living at the age a:, and 4j_i the number 
of those that will be living at the age x\\^ and so for other 
ages. 

The amount v^ at the rate of interest r, will be what it 
is now worth to insure one dollar, to be paid certain at the 
end of one year; and ?;" is the amount that will insure one 
dollar, to be paid certain at the end of n years. 

To insure one dollar, to be paid at the end of one year 
to the heirs of a person in case he dies during the year, it 
is necessary to multiply the amount that would, at a certain 
rate of interest, produce one dollar certain, at the end of 
one year, by the fraction which expresses the chance or 
probability that the insured will die during the year. 

The amount that will produce one dollar certain at the 
end of one year is v. The fraction representing the chance 
of the insured d3'ing during the year is expressed by the 
number of deaths during the year, divided by the number 
living at the beginning of the year. The number of deaths 
during the year is equal to the number living at the begin- 
ning of the year, minus the number living at the end of the 
year. The number of deaths, therefore, between the age x 
and the age x\-\ is expressed by 4 — 4^i; and the fraction 
expressing the chance or probability that the insured will 
die during the year, between the age x and the age :c-|-l, is 

-i^^|±i. This, multiplied by v^ gives v -"'"^^^ for the amount 

that will insure one dollar, to be paid to the heirs of a 
person aged x^ in case he dies during the first year follow- 
ing the date of the transaction. In like manner we can 



Life Insurance, 4^ 

determine what it is now worth to insure the person against 
d^dng during the second year, the third year, and any year — 
and every year, up to and including the table limit of 
the duration of human life ; and by adding together these 
respective values for every year, we obtain what it is now 
worth to insure the one dollar for life. That is to say, 
we obtain the amount that will enable the Company to 
insure one dollar to be paid to the heirs of the insured at 
the end of any year in which he may die. Calling this 
amount or net single premium that will insure one dollar 
for life, at the age x, ^P^,, we have — 

s-?=v^'-^^-{-v'l' +^~\'+' + t)^^-f±i^i'+i+, &c., to age 99; 

^X *X ''X 

^3! ''X 

&c. 

Now let us obtain an expression for the present value, or 
sum in hand, that will, at any age x, be the precise money 
equivalent of a future life series of annual premiums or 
payments of one dollar ; the first payment to be made in 
advance, and one at the beginning of each year following, 
provided the insured is alive to make the payment. The 
first payment, and therefore the first term of the series, is 
one dollar. This is paid in hand; is certain; is equal to 

unity or one dollar, and may be represented by ~. The 

present value of one dollar, to be paid certain at the end 
of one year, is v, but the second payment, at the end of 
the first or beginning of the second year, is only to be 
made in case the insured is alive at the time the second 
payment is due. The chance or probability that the in- 
sured will be alive is obtained by dividing the number 
living at the age x-\-\ by the number living at the age x. 
Therefore, the present value of the second payment is ex- 
pressed by v~^. By similar reasoning, the present value of 

the third payment is i;^ -yt2 . and so for the other payments. 



46 Jfotes on 

Calling the value of all the payments A^., we are at once 
enabled to write the equation : 

A^= - -j-t;f±}-|-^;- jl±i -]-, &c., to ninety-nine years of age, 

''X f-x f-x 

(2) or A,,= ^ ^+— '--^^ ' to ninety-mne. 

''X 

By combining equation (2) with equation (1) we have : 

The expression A^, gives us the amount in hand, which is 
the precise money equivalent of a future life series of an- 
nual premiums of one dollar at the age x. Therefore A^ is 
to 5?^, as a future life series of annual premiums of one 
dollar is to a future life series of annual premiums that 
will be the precise money equivalent of the amount sY^ in 
hand ; from which we deduce the annual premium which 

is the equivalent of ^P^ in hand : it is — -^; and as ^P.^ in hand 

will insure one dollar for life at the age x^ therefore its 
equivalent, in money value — an annual premium amounting 

sP 

to — - : will insure one dollar for life at the age x. Calling 

A, 
this net annual premium «P^, we have : 

(4)«P='-g=^-(l-t,). 

The net annual premium at any age x being «P^ ; that at 
age x-\-n will be aP^+n? ^-^d the net single premium, at age^ 
.r-f-Ti, will be ^P,;^„. The value of a policy at the age x-\-n^ 
or the net single premium that will purchase a whole life 
policy of one dollar at the age x\n^ is ^P^.^^. The value 
at age X'\-n of a future life series of annual premiums, each 
equal «P.^, is represented by «P^XAa._)_„. 

The " Deposit " at the end of the in}^ year of a whole life 
policy, taken out at the age x, represented by (T.F.D.)x-|-7i, 
must be sufficient to make, at the age x-\^n^ when added to 
the value, at that age, of the future life series of a^j. net 
annual premiums, an amount equal to the net single pre- 
mium at that age. We therefore have : 

.P,+,-aP.XA,+„=(T. F. V,)x\n; 



\ 



Life Insurance. Jj, 7 

From equation (3) we deduce sV^j^,==\ — (1 — -y )A^^_„. 
Therefore (T. F. D.).+.-l-(l-t')A.+.-aP.XA.+.. 

But a?,=^^= i- —{\—v.) See equation (4.) There- 
fore, (T. F. D.).+„= 1 - (l-«) A,+.-^-l- -(l-t,)\ A,+„ ; 

(5) or (T. F. D.U„=l-^-±^. 

We have previously shown how to obtain the value of 
A at any age. The fraction expressing the value is found 
by taking the numerator and denominator in the N^ and 
D^ columns of the table, opposite the designated age. 
Having obtained by this means the value of A, the net 
single premium (see equation 3) is expressed by: 

The net annual premium <2P^=---- — (1 — v.) (See eq. 4.) 
And the '' Deposit" (T. F. D ),+„-=!— ^^. (See eq. 5.) 

■^x 

Having obtained the requisite "Deposit" that must be 
on hand at the end of any year ; by subtracting this from 
the amount of the policy, we have the amount at risk 
during that year. Multiply the amount at risk during any 
year by the fraction which represents the chance or prob- 
ability that the insured will die during the year, and the 
result will give the " Cost of Insurance " during that year. 

The net annual premium at net interest for the first year 
will be exactly sufficient to pay the net Cost of Insurance 
during the year, and provide the "Deposit" for the end of 
the year. 

The net annual premium paid at the beginning of the 
second year, added to the "Deposit" at the end of the 
first year, will produce a sum which, increased b}" net in- 
terest, will exactly pay the "Cost of Insurance" during the 
second year, and provide for the "Deposit" that must be 
on hand at the end of the second year; and in like manner 
for the third year, fourth, and every year up to, and includ- 
ing, the limit of the duration of human life, according to 
the law of mortality expressed by the table used. 



48 Js^otes on 

Thus the net annual premium paid the last year of the 
table will, when added to the "Deposit" for the end of the 
preceding year, make a sum which, at net interest for one 
year, will exactl}" amount to the face of the policy, and 
this, too, after paying " Cost of Insurance," properly charge- 
able to this policy in every previous year. 

From what precedes, it is seen that at any age x^ the 
amount is easily calculated that will, if paid in hand, in- 
sure one dollar to be paid to the heirs of a person at the 
end of any number of years indicated by %, in case the 
insured dies during the n^^^ year. By making this calcula- 
tion for every year, to the limit of the Table of Mortality, 
and adding together all these respective yearly results, we 
obtain the amount that will, if paid in hand, insure the 
one dollar to be paid to the heirs at the end of any year 
in which the insured may die. 

The amount that will insure one dollar, in case the in- 
sured dies during the first year, is very small when x is 
small. For instance, at age twenty, it requires, to insure 
one dollar, between the age twenty and the age twenty- 
one, an amount equal to v^ multiplied by the fraction which 
expresses the chance or probability that the insured may die 

during the year; or, $0.961538X-^^^= $0.0070104. For 
•^ 93,268 

a policy of $1,000 the amount required is $7.01. This is 

cheap for the first year; but must be paid in cash in 

advance. Bear in mind that this method provides no 

"Deposit" for the payment of policies that mature in the 

future. 

The net annual premium at the same age, to insure $1,000 

for whole life, is $12.95; but this amount will, in addition to 

paying the "Cost of Insurance" year by year, furnish the 

means for providing the requisite " Deposit," and make the 

ultimate payment of the policy at maturity certain So 

that at age ninety-nine, in case the insured is still living, 

he will, at that age, only have to pay $12.95 to effect his 

insurance during the one hundredth year of his age. Let 

us see how the matter will stand at age ninety-nine with 

a person who has no "Deposit" accumulated from year to 

year. At age ninety-nine the amount requisite to insure 



Life Insurance, 4^ 

one dollar for one year, is v multiplied by the fraction 
which expresses the chance or probability that the insured 
will die during the one hundredth year of his age ; or, 

vX—==v= $0.961538. To insure $1,000 for one year at the 

same age will require $961.54. 
4 



50 Jfotes on 

riNSURANCE OTHER THAN WHOLE LIFE. 

Value of n Annual Payments of One Dollar Each. 
We have previously seen that: 

_v''l^-\-v''-^\_^^-\-v''^^-l^j^2-\- •) &^c., to age ninety-nine; 

and that this expression assumed the form : 

. _ D.+D.+i+D»^.+ ■ . ■ . to D,, _N, 

^' d: D-; 

Suppose that it is desired to find an expressioh for the 
present value of a series of n annual payments of one dol- 
lar each, provided the person is alive to make the payments, 
the first being made in advance at the age x. 

It is clear that if we take the first n terms of the second 
member in either of the two equations above, and make 
the calculations indicated, we will obtain the desired 
result. But by using the N^ and D.^ columns of the table, 
the arithmetical operation can be very much shortened. 
We find the value of N,. from the table by taking the num- 
ber in that column opposite the age x. In doing this, we 
have taken the whole series of annual payments of one 
dollar to the limit, ninety-nine years ; but this is too much, 
because we want only the first n terms. We have, there- 
fore, to find the value of the terms not included in the first 
n terms, and subtract their sum from the sum of all the 
terms in N,., which will give us the present value of the 
first n terms. The first payment is made in hand at the 
age x; the second is made at the age :c-t-l, and n pay- 
ments will have been made at the age x-^n — 1. All the 
terms following this last are to be subtracted from the N^ 
series; that is to say, we commence at the age x^n and 
take the series from that term to the limit, ninety-nine 
years, and subtract the sum of this series of terms from 
that obtained by starting at the age x, and taking the series 
through to age ninety-nine. The latter is expressed by JM^. ; 
the former by N<.^„. Therefore, the present value of one 
dollar, payable annually for n years, provided the person 
is alive to pay it — the first payment being made in ad- 



Life Insurance, 51 

vance ; the age of the person being x years — is expressed 
by -^ ^+ -. Calling this A,L we have AJ.='-^^-^\ 
The symbol A^L indicates that the first n payments only of 
one dollar are taken : 

Example. 

At age thirty^ what is the value of a series of twenty annual 
payments of one dollar each, the first payment to be made in 
advance, and one at the beginning of each succeeding year, pro- 
vided the person is alive to make the payment ? 



'-A-a;L 



D, 



x=ZO, n=20. 



inen ^ - — . 

N3o=479951.6. N5o=131765.6. 



N30— N5o=348186.0. D3o=26,605.37. 
$13.087057. 



N30— N50 348186.0 



D30 26605.37 

Value of a Life Series of Annual Payments of One Dol- 
lar Each, the First Payment to be Made at the End op 
n Years. 

The symbol nA^ is used to represent that the payments of 
the one dollar are postponed n years, and then commence 
and continue for life. In this case we have only to omit 
the first n terms of the numerators in the expression for A^, 

and we have : nA^= —^' 

Example. 

What is the value, at age forty, of a life series of annual 
payments of one dollar each, the first payment to be made at 
the end of thirty years, and continue for life ? 

N 
nA^=-"i-. x=40. w=30. 



5'^ J^otes on 

Term Insurance. 

Let us suppose it is desired to find an expression for the 
net single premium that will insure one dollar for a term 
of years only. Let the number of years, as before, be 
represented by n; we have before found 

,P _,/x+.<+i+^' 44-2+ _ ^^■^+1+ ^'' 4+ 2+^' 4^3 +> &c » 

^' ^ . . . ^ J^ 

Multiplying both the numerator and denominator of each 
of the above fractions by v"" , we have — 



sV.= v 



^-7,+^^+^ 4+1+^^+' 4+2+ 



which becomes, 



^«+i4+i+t.--^^4+,+^-+=/+3.4-, &c 



as the series in this case extends to ninety-nine years : 



r,N. N.+1 



Observe that the first term in the series, contained in the 
numerator of the negative term of the second member of 
the above equation, begins with the year x-^l. 

The above expression for ^P^. is the net single premium 
that will insure one dollar for life; but we want to find an 
expression for the first n years only. We must, therefore, 
subtract from the above expression the value of all the 
terms of this .series ^ter passing the first 7i terms. Leav- 
ing out these first n terms, the sum of the series, from age 

N 
x-\-n to age ninety-nine, will be expressed by '^ --^^ — 

^ r. 



N,+„+i 



and subtracting this from the whole series for life, 



as given above, and calling the net single premium that 
will insure one dollar for n years at age a:, s?A^ we have: 

Or, .P I = KN-N,^„)-(N,+,-N,^„^0 

This gives the value of the net single premium that will 
insure one dollar for a term of n years. 



Life Insurance. 53 

Example. 

What net single premium is required at age thirty to insure 
^1,000 for twenty years ? 

H " ~' D 

N^=N3o=479951.6. N,+„=N5o=131765.6. 
N30— N5o=348186.0. V (N30— N^o) -=334794.07. 
N,+i=N3i=453346.2. N,^.„+i=N5i=121983.7. 
N31— N5i=331362.5. 
2;(N3o-N5o)-(N3 -N5i)=:3431.57. D,=-D3o-26605.37. 

Therefore sV±=sVJ,,= J-gl-^Z- ==^0.1289803. 
j I 2Db05.o7 

This will insure one dollar as above. Multiply by 1,000, 
and we have the result required — $128.98. 

Note. — In case the net single premium has been calculated as 
above ^ and the numerical value of A^-L corresponding thereto y 

has been determined; the net annual premium is obtained by 
dividing ^P^L by A, 



Net Annual Premium for the Above Policy. 

The amount A-,L in hand is the equivalent of an annual 

premium of one dollar for n years, therefore sVA^ in hand 

is the equivalent of a proportional annual premium for n 
years. Hence — 

-A-xL * *Px L : ^ $1 : A I ■• 

This fourth term is an expression for the net annual 
premium that will insure one dollar for n years. By sub- 
stituting for ^P^L and A^L, in this expression, their values, as 

obtained above, we have — 

sVX _ ^(N -N,^„)-(N,^,-N.^.^O N.^,-N,^..^, . 

AJ. N -N.+„ N.,-N.,+„ • 

This is the formula that is used in calculating the net 

annual premium that will insure one dollar for a term of 

n years. 



5Ji> J^otes on 

Example. 

What is the net annual premium that will, at age thirty, insure 
$1,000 for twenty years? 

Aj; N.-N.+. N30-N30' 

N31— N5i=331362.5. N30— N5o=348186.0. 
N31— N51 331362.5 



N30— N50 348186.0 

^. N3,-N,_ 



$0.951682. 



N3o-N,o 



.961538 — $0.951682=$0.009856. 



This is the net annual premium that will insure one 
dollar as above. Multiply by 1,000, and we have $9,856, 
which is the net annual premium that will insure $1,000 
as above. 

Endowment. 
By an " Endowment," money is assured to be paid in 
case the person insured is living at the time named in the 
endowment policy, say n years from its date. The present 
probability of his being alive at the age x-\-n is equal to 
the fraction obtained by dividing the number of those living 
at the age x-\rn by the number living at the age x. This 
fraction multiplied by the present value of one dollar, to be 
paid certain at the end of n years, will give the present 

value of an endowment of one dollar, that is : •*'+" . But 

this is equal to - — i^±!i=:±^. 

Representing the net single premium for an endowment 
of one dollar at age x, payable in n years, by E^|„, we have 

the equation EJ„= -^. 

Example. 

What is the net single premium that will, at age thirty, insure 
an endowment of $1,000, to be paid at age fifty? 

Ej„=^i!i=5^=J^Z?i:??=$0.367667. 
D^ D,n 26605.37 



Life Insurance.^ 55 

This is the net single premium that will insure one dollar. 
Multiply by 1,000, and we have $367.66, which is the net 
single premium that will, at age thirty, insure an endow- 
ment of $1,000, to be paid at age fifty, in case the insured 
is alive at that time. 

Net Annual Premium for this Endowment. 

A^|„ in hand is the equivalent of an annual premium of 
one dollar for n years, and the net annual premium in this 

F I 
case is expressed by . '^ ■. Substituting in this expression 

for E,L its value, t^^ and for A,L its value, ^ '"+" , 



the expression becomes 



^x\n ^x+n 



A I N N , 



This is the formula used for calculating the net annual 
premium for an endowment to be paid at the end of n 
years. 

Example. 

What is the net annual premium that will insure an Endow- 
ment of $1,000 as above? 



EJ._ D.+. _ D50 _ 9781.92 



:$0.028094. 



AJ. N.— N,_^, N30— N50 348186.0 
This will insure one dollar as above. Multiply by 
1,000, and we have $28.09, which is the net annual pre- 
mium at age thirty for an endowment of $1,000, to be paid 
in twenty years, or at age fifty — in case the insured is 
alive at that time to receive the endowment. 

Endowment and Term Insurance Combined. 

In case of endowment, payable at age x^n^ and insur- 
ance payable at death if previous, the net single premium 
for the endowment is added to the net single premium for 
the term insurance, and we have the expression : 5PJ,^+Ej.L. 

Substituting for each of these expressions their respective 



56 J^otes on 

values as given above, and observing that Da.^.„+Na._|. 
is equivalent to Nj._,_„, we have the equation : 



«+l 



Example. 
What is the net single premium that will^ at age thirty, insure 
$1,000 to be paid in case the insured dies within twenty years, 
and at the same time insure an endowment of $1,000, to he paid 
at the end of twenty years^ in case the insured is alive at that 
time ? 
(,T> vw N.-.„+t>(N-N,^„)-N.+i . N,„+t>(N3,-N,o)-N3i 

(^p+E).| = ^ — D^ = d;^ • 

t;(N3o— N5o)=334794.07 ; add N50 and we have : 
334794.074-131765.6=466559.67; subtract 
N31 from this and we have : 13213.47 ; and 

N,o+.(N3o-N.)-N3 .^ 13213.47 ^^0 ,36^46. 
D30 26605.37 

Multiply this result by $1,000, and it gives $496,65, which 
is the required net single premium. 

Net Annual Premium for the Above. 

The net annual premium is equal to ^ — - — ^— ; and by 

substituting for the numerator and denominator their re- 
spective equivalent expressions, as given above, we have 

V — ^^ ""^^ This is the formula used for calculating 

the net annual premium for a policy combining Endow- 
ment and Insurance of one dollar, payable in n years, or 
at death, if prior. This is usually called "Endowment/* 
simply. 

Example. 

To find the corresponding net annual premium. 
(^P+E)Jn _^. N..^ -N.^ N3,-N,o _^, 321580.6 _ 

AJ. N-N.+n N30-N30 848186.0 

V— $0.923588=$0.961538— $0.923588=$0.037950. Multiply 
this by 1,000, and we have $37.95, the required net annual 
premium. 



Life Insurance. 57 

Whole Life Insurance Paid for in n Years. 

To find the formula used in calculating the net annual 
premium for whole life insurance, payable by n annual pre- 
miums, we use again the proportion : as A^L, in hand, is 

the equivalent of n annual payments of one* dollar, any 
other amount, in hand, would be the equivalent of a pro- 
portional annual payment for n years. We have already 
seen that ^P^, in hand, is the net single premium that will 
insure one dollar for life — hence, the proportion : 

AJ„ : .P. : : $1 : f^. 

The fourth term of this proportion is the net annual pre- 
mium for n years that will be equivalent to sV-, in hand; 
but ^P^ in hand is the net single premium that will insure 

^P 
one dollar for life ; therefore, — j- is the net annual pre- 

mium for n years that will insure one dollar for life. It 
has before been seen that5Pa.= l — (I — ij)A^=1 — (1 — v) 

"^ — "^ ^ ""' and A^L is equal to "^ '^^"• 



therefore, 4—^= ""Z^ T^ ^ '' » This is the formula used 

for calculating the net annual premium for n years that 
will insure one dollar to be paid to the heirs of the policy- 
holder at the end of the year in which he may die. 

Example. 
What is the net annual premium for ten years, at age 
forty, that will insure $1,000 for life? 
^P._ P.— (l-i;)N, _ D,,— (1— t;)N,o 
Ai 1^ — i^x+n N,o— N,o • 

D,o=16382.56; (1— v)=$0.038462 ; N,o=263643.5. 
N5o=131765.6. From which we obtain — 

D40— (1—^)^40 _ 1638256—1014025629 _ 6242.30371 ^ 

N^,^— N50 263643.5—131765.6 13 J 877.9 " 

0.04733. Multiply by 1,000, and we have $47.33, the net 

annual premium required to insure $1,000 for life, at a^e 

forty, in ten annual premiums. 



S8 Jfotes on 



GENERAL FORMULA FOR THE TRUST FUND DEPOSIT. 

The net annual premium may be considered as composed 
of two separate and distinct parts, each part accurately 
adjusted to accomplish a specific purpose. One portion of 
the net annual premium is intended to pay losses that will 
occur each year by the death of a certain number of policy- 
holders, as indicated by the Table of Mortality ; the other 
portion of the net annual premium is intended to provide 
the amount that the Company must hold, at the end of 
each year, on deposit, in trust for the policy-holder. Of 
course, in order to accomplish these purposes, each of the 
respective parts or portions into which we assume that the 
net annual premium may be divided, must be increased by 
net interest. 

The value of the risk for one year, at age x, on one dol- 
lar, has been shown to be v ~^. That is to say, v ~ will, 

when increased by net interest for one year, be the amount 
that will pay the risk on one dollar. The net annual pre- 
mium is more than the actual value of the risk on one 
dollar for the year. This excess in the payment makes the 
actual "Amount at Risk" always less than the amount of 
the policy. The " Amount at Risk " during the first year, 
on a policy of one dollar, has been shown to be equal to 
$1-(T. F. D.).+i. 

The value of the risk on one dollar being v -^, the actual 

risk on the amount 81 — (T. F. D.),-^i, is expressed by 

2>-^ 1 — (T. F. D.),^i.| Subtract this from the net annual 

premium, and we have that portion of the net annual pre- 
mium that is intended to provide the requisite amount to 

be held on deposit in trust, viz: flP^ — "^T (-^ — C^' ^' ^Ox+i) 

Add to this net interest for one year, and we have the 
amount that must be held on deposit in trust for the policy- 
holder at the end of the first year of the policy. 



Life Insurance. 59 

Let the ^^ ratio ^'' of interest be represented by r. Observe 
that this r is not the rate of interest; it is a quantity which 
will, when the principal is multiplied by this quantity, pro- 
duce what the principal will amount to when increased by 
net interest for one year. For instance, v being the prin- 
cipal and r the ratio of interest, rv is equal to one dollar; 
and of course r is equal to one divided hy v. 

All that is now to be done in order to form an equation 
that will enable us to determine the amount that must be 
held by the Company at the end of the first year on deposit 
in trust for the policy-holder, is to write the expression we 
use to represent this amount, viz : (T. F. D.)^^.!, and place 
it equal to what that portion of the net annual premium 
not required to pay losses by death, viz : 

will become, when net interest for one year is added to it. 
This amount, resulting from adding the interest to the 
principal, is obtained by multiplying the principal by the 
^ ratio^^ of interest, which is r; therefore — 

(T. F. D.).+i=r(«P,-t>^(l-(T.F.D.).+,.) 

This is a simple equation of the first degree, containing 
only one unknown quantity, and that is the quantity sought 
for, viz : the amount on deposit in trust at the end of the 
first year. 

It is true the unknown quantity is found in both members 
of the equation ; but that is usual in the simplest elementary 
problems in algebra. We have used the above notation 
because it was supposed to be easy for the general reader 
to retain the idea that (T. F. D.).,._j_i means "//i^? amount the 
Company must hold, on deposit, in trust for the policy-holder, at 
the end of the first year of a policy taken out at the age .t." 

Suppose we had called this amount y, and had repre- 
sented the net annual premium that will insure one dollar 
for life at the age x, by p ; and that the present value of 
one dollar, to be paid certain at the end of one year, mul- 
tiplied by the fraction which represents the chance that the 
policy-holder will die during the first year, is represented 



60 Jfotes on 

by h. Then r, p, and h are known quantities, and the 
numerical value of each is easily determined. 

The above equation, when written with the symbols just 
assumed, will become, y'=r{p — 6(1 — y))\ or y=^rp — rb\- 
vhy; ov y — rhy^=ri^p — h)\ or?/(l — rb)=^r{p — h)\ hence j/= 

(p — o). As these quantities r, p, and b, are easily 

calculated, and their arithmetical value readily determined? 
it is clear that there can be no very abstruse mathematics 
required in the solution of this question. ■ 

This digression was entered upon for the purpose of illus- 
trating the fact, that no very " high order of mathematical 
attainment'''' is essential to a clear comprehension of the 
principles and formula used in Life Insurance calculations. 

We will resume the subject of the "amount that must 
be on deposit, in trust, at the end of the first year," by 
again writing the general equation : 

(T. F. D.).+i = r(aP,-t.^'(l-(T. F. D.).+o) 
Call V ~, <?_^, and the equation becomes, 

(T. F. D.Ui = rfaP.-c,(l-(T.F.D.UO^; 



or (T. F. D.Ui=raP,-rc,-f rc,(T. F. D-U^; 
or (T.F. D.),+i-rc,(T. F. D.^i^raP,- re, ; 
or (l-rc,)(T. F. D .).^i = r («P, — c,) ; 

hence (T. F. D.)^^,^--^ (^P^-O- 

1 — rc^ 

V 

Call M^, and we have, 

1 — rc^ 

(T. F. D.Ui=7^,(«P.-c,.) 

This^ is the formula used in calculating the amount that 
should be held by a Life Insurance Company on deposit in 
trust at the end of the first year of a policy taken out at 
the age x. 

To obtain the formula used in calculating the amount on 
deposit at the end of two years, we first find the value of 



Life Insurance. 61 

the risk on one dollar during the second year. This is 

expressed by v y^^, or c^_^i. The actual " Amount at Risk" 

during the second year is equal to 1 — (T. F. D.)a.^_2. 
Therefore, the actual value of the real risk during the 

second year, is c^^Il — (T. F. D.)^,2- ) Taking this from 



aP^, we obtain that portion of the net annual premium paid 
the second year, that is applied towards forming the amount 
that must be held in deposit at the end of that year. Bear 
in mind that the amount in deposit at the end of the first 
year is exclusively used in making up the amount that 
must be held in deposit at the end of the second year. 
We are, therefore, enabled to write the equation (T. F. 



D.)x+2 = H(T. F. D.).+i+aP.-c.+i(l-(T. F. D.).+2.) 
B}'" transposing as before, and finding the value of the 
unknown quantity (T. F. D.)^^25 and calling — 
^x+i) we obtain the following equation : 

(T. F. D.),+, = «,+,(( T. F. D.).+i+ aV,-c, 



rc 



x4-l 



By similar reasoning we are enabled to write out the 
general expression or formula for obtaining the amount 
that must be, by the Life Insurance Company, held on 
deposit, in trust for the policy-holder, at the end of n years 
from the date of his policy, taken out at the age x. It is 
as follows : 



(T. F. D.),^.»=«,+„_j^(T.F. D.U„_,+«P-o.+„_ 

This formula applies in calculating the amount that must 

56 held in deposit at the end of n years from the date of 

t^he policy ; and is general, being applicable to all kinds of 

Dolicies. The formula deduced previously is used only in 

letermining the amount on deposit for whole life policies. 

t will be noticed that, by the latter, the calculations can be 

aade for any named year, without reference to the amount 

•n deposit any previous year. The general formula deduced 

jbove requires that the amount on deposit at the end of the 

ear just preceding should be known, before the amount for 

be end of the year in question can be calculated. 



62 JVotes on 

Tables have been constructed containing the values of ux 
and cz at the different ages. 

Example. 

What ts the amount that must be held by the Company 
on deposit, in trust for the policy-holder, at the end of 
the first year, on a twenty years' "endowment and insur- 
ance" policy for $1,000, taken out at age thirty? 

(T. F. D.).^+i = i^^(«P^ — c^.) As before calculated, aV-,= 
$0.03795. From the table ?z,= 1.04884, and c^ = 0.008101. 
Therefore (T. F. D.),.+i = $0.03131. Multiply by 1,000, we 
have $31.31, which is the amount that must be on deposit 
at the end of the first year on the above policy. 

To Determine the Amount that Must be Held by the 

Company at the End of the Second Year on Deposit, in 

Trust for the Policy-holder, on the Above Endowment 

AND Insurance Policy. 

We have, as before, «P,=$0. 03795 : (T. F. D.),+i, as 
above calculated, is $0.03131 ; 2^^^^= 1.04900 from the table; 
and c^.^.i=0. 008248 ; this is. also taken from the table. 

We substitute these numerical values in the formula, and 
it becomes — 

(T. F. D.)^^2=l 04900 ($0.0313l4-$0.03795— $0.008248). 

Performing the arithmetical operations indicated, we have 
(T. F. D.),4.2=$0. 64001588; multiply by 1,000, and we have 
$64.00. This is the amount that must be in deposit for the 
above policy at the end of the second policy year. 

Amount of the Deposit in Case the Policy is Paid for by 
A Net Single Premium. 
In case a policy is paid for by a net single premium, at 
any age x, at the end of the first policy year, the Company 
must have on hand, deposited to the credit of the policy, 
after paying the not cost of insurance during the year, an 
amount that will be sufficient to make the payment of the 
policy at maturity safe. In other words, the amount on 
hand in deposit, at the end of the first policy year, must be 
equal to the net single premium at the age x+1 ; and, in 
like manner, the deposit at the end of n years must be 
equal to the net single premium for the year x-{-n. 



Life Insurance. 63 



ANOTHER FORMULA FOR THE AMOUNT ON DEPOSIT 

May be obtained as follows : The number of persons living, 
as given in the Mortality Table, at the age x-^n, is expressed 
by ljc+7i' Suppose all these persons are insured for one dollar 
each. What is the amount on deposit for each policy-holder 
living at the end of the year x-\'n-\-\ ? 

The amount on deposit for each policy-holder, at the end 
of n years, is expressed by (T. F. D.)^^^. At the beginning 
of the n^^^l year, each policy-holder pays his net annual 
premium, represented by a?^. To obtain the net funds of 
this Company for the year, we add the net annual premium 
to the amount on deposit; multiply this sum by the number 
of policy-holders living, and add net interest for one year. 

The ''ratio'' of interest is r; therefore, /(T. F. D.),_j.^4- 

«Px jylx-^nxT is an expression for the whole net funds, with 

interest for the year added. Out of this the cost of insur- 
ance for the year must be paid. The number of deaths 
during the year is d-^j^^. Each policy is one dollar. There- 
fore d^_^n represents the cost of insurance for the year in 
question. Subtract this from the net funds on hand, and 
we have the whole amount that the Company must hold on 
deposit in trust for its policy-holders at the end of n^l 
years. This amount on deposit for each policy-holder is 
expressed by (T. F. D.)^+„+i. The number of policy-holders 
living at the age x-\-n-\-l, is expressed by 4^.^_[.i. Therefore, 



or 



4+,.+, (T. F. D.) .+.+1=4+,. ((T. F. D.).+„+aP,jr-4+„ ; 
(T. F. D.).+„+i=r^((T, F. D.),+,.+aP.)-^. 



It will be noticed that this formula, too, requires that, 
before the deposit at the end of any policy year can be 
calculated, the deposit at the end of the next preceding 
year must be known. All the other terms of the second 
member of the equation are known quantities. 

To determine, by this formula, the amount that must be 
in deposit at the end of the llrst year on a twenty years' 



64 JSFotes on 

endowment and insurance policy for $1,000, taken out at 
age thirty. 

In this case a:=30 and n=^0. The formula then becomes : 

(T. F. D.Ui=r-^((T. F. D.).+«Px)-v^; 

but rv=\ ; and r= —=1.04. From the Mortality Table we 

find Z3o=86292, and /3i=85565. (T. F. D.)^ is equal to zero, 
because there is no deposit at the end of the year preceding 
the first policy year. «P^, the net annual premium, is 
$0.03795, as previously calculated ; and d^^ by the table is 
727. 

Substituting these arithmetical amounts in the above for- 
mula and it becomes — 

(T. F. D.)3i=1.04X |?^X 0.03795 ^^^ 



85565 ' 85565' 

or (T. F. D.)3i=S0.031307. Multiply this by 1,000, we have 
$31.31 for the amount of the deposit at the end of the first 
year. The precise figures are $31,307, which is thirty-one 
dollars, thirty cents, and seven mills, or $31.31. 



Life Insurance. 65 

METHOD OF CALCULATING THE NET VALUE OF A POLICY 
FOR FRACTIONAL PARTS OF ANY POLICY YEAR. 

At the time the tirst premium is paid, which is at the 
beginning of the first policy year, the net value of the policy 
is the net annual premium. At the end of the first policy 
year the net cost of insurance will have been paid, and 
there must be left in the hands of the Company, in trust 
for the policy-holder, the requisite " Deposit." This Deposit 
(or " Reserve,''^ as it is often called) is the net value of the 
policy at the end of the first policy year. At the beginning 
of the second policy year the net annual premium is paid, 
and the net value of the policy is then the " Deposit" at the 
end of the preceding year, plus the net annual premium just 
paid. The net value of the policy at the end of the second 
policy year is the Deposit (or " Reserve ") for the end of that 
year. 

In general term^, the net value of the policy, at the begin- 
ning of the n*^ policy year, is equal to the " Deposit " at the 
end of the n*^ — 1 policy year, plus the net annual premium ; 
and the net value of the policy, at the end of the n^^ policy 
year, is equal to the " Deposit" at the end of n years. 

This is true, because the net annual premium is sufficient, 
and only sufficient, when added to the "Deposit" at the 
end of the preceding year, to pay the net cost of insurance 
during the year, and provide the requisite deposit for the 
end of the year. Of course it is understood that net or 
table interest is realized for the year. 

On the supposition that a policy was taken out on the 
1st day of January, 1869, the net value of the policy on 
that day, is equal to the net annual premium just paid. 
On the 31st December, 1869, the net value is equal to the 
''Deposit" at the end of the first policy year. On the 1st 
day of January, 1870, the net value of the policy, just after 
the net annual premium is paid, is equal to the Deposit at 
the end of the preceding year, plus the net annual pre- 
mium; and the net value on the 31st of December, 1870, 
will be equal to the Deposit at the end of the second policy 
year. 

5 



66 J{otes on 

Having in this way determined the value of this policy 
at the beginning and at the end of any policy year : sub- 
tract one from the other, and by this means obtain the 
difference between the net value on the 1st day of January 
and the net value on the 31st day of December of that year. 
Divide this difference by twelve : we will obtain the monthly 
difference in the net value. Assuming that the net value of 
a policy is greater at the beginning, than it is at the end of 
the policy year in question ; having found the monthly differ- 
ence as above, we will subtract this monthly difference from 
the net value at the beginning of the year, in order to find 
the net value of this policy on the 1st day of February of 
that policy year. To find the net value of the policy on 
the 1st day of March, we will subtract the monthly differ- 
ence from the net value on the 1st day of February: and 
in like manner we obtain the net value of the policy at the 
beginning of any mont i of the policy year, by subtracting 
from the net value at the beginning of the policy year, this 
monthly difference, multiplied by the number of months of 
the policy year that have expired. 

On the 1st day of November, for instance, we obtain the 
net value by multiplying the monthly difference by ten, and 
subtracting the result from the net value of the policy on 
the 1st day of January, which day we have assumed to be, 
in this case, the first day of the policy year. 

To obtain the net value on any day during a month, divide 
the monthly difference by thirty, in order to obtain the daily 
difference ; and then use the daily difference in a manner 
entirely similar to that indicated above for finding the value 
of the policy at the beginning of any month. 

Policies are taken out any day of the year, and it is usual 
in Life Insurance Companies to have the net valuation of 
all policies computed on some one day every year. The 
day fixed for these valuations is generally the 31st of 
December. 

The question will then arise every year, what is the net 
value, on the 31st of December, of each policy in force on 
that day ? 

First determine what policy year the given policy is in at 
the time. Obtain its net value at the beginning of that 



Life Insurance. 67 

policy year, and its net value at the end of that policy 
year. Take the difference between these two net values : 
divide this difference by twelve, in order to obtain the 
monthly difference in the net value; divide the monthly 
difference by thirty, in order to obtain the daily difference 
in net value. Then fix the month and day of the calen- 
dar year on which the policy was issued. The number of 
months and days that have, on the 31st of December, 
elapsed since the beginning of the policy year, will become 
known, and the net value of the policy on the 31st day of 
December can be determined by the general method above 
indicated. 

Value at the End of a Policy Year is Sometimes Greater 
THAN the Value at the Beginning. 

Owing to some peculiarity in the rate of mortality for 
the year, and the accumulation of interest arising from the 
funds on deposit, it happens at times that the net value of 
a policy, at the beginning of a year, will, at net interest, 
produce, during the year, an amount sufficient to pay the 
cost of insurance during the year, and provide for a "De- 
posit " at the end of the policy year, greater than the net 
value at the beginning of the year. In this case, the 
monthly and daily differences must be added to the net 
value at the beginning of the year, instead of being sub- 
tracted from it. This peculiar case does not happen in 
the earlier years of a policy; it is only after there is marked 
accumulation in the "Deposit" or net value at the end of 
a year, that the net value at the beginning of a year will, 
at net interest, produce an amount sufficient to pay the 
cost of insurance during the year, and leave on hand at 
the end of the year a " Deposit," or net value, greater than 
that at the beginning of the year. 

These ^'■perturbations'''' in the relative net values at the 
beginning and end of different years are indicated in the 
formula by unmistakable signs ; they, in no degree, com- 
plicate the calculations, but require close observation on 
the part of computers to prevent mistakes. 



68 J^otes on 

To Find the Net Value During the Year by Usikg Net 
Value at the End of the Year. 

The net value, at any time during a policy year, can be 
obtained with equal certainty by basing the calculations 
upon the Deposit or net value of the policy at the end of 
the policy year, instead of, as above, upon the net value at 
the beginning of the policy year. 

Having calculated the Deposit that must be on hand at 
the end of the policy year, the value of the policy at the 
end of the first month of the policy year may be obtained 
by adding to the Deposit that must be on hand at the end 
of the year eleven twelfths of the cost of insurance during 
the year. At the end of the second month the net value 
may be obtained by adding to the Deposit that must be on 
hand at the end of the year ten twelfths of the cost of 
insurance during the year. At the end of the eleventh 
month one twelfth is added. At the end of the twelfth 
month, or end of the policy year, there is nothing to be 
added. The net value, and the Deposit at the end of the 
year, are equal quantities. 

What is said above in reference to the particular case 
in which the Deposit or net value at the end of a policy 
year is greater than the net value at the beginning of the 
year, applies here ; and, therefore, when the case occurs, 
the eleven twelfths of the difference between the net value 
at the beginning and that at the end of the year, must be 
subtracted from the net value or Deposit at the end of the 
year, in order to obtain the net value at the end of the 
first month of the policy year; and in like manner for other 
months. 

It is assumed in both of the methods for calculating the 
net value of a policy during the policy year, that the varia- 
tion in value is proportional to the time ; and that each 
month has thirty days. 



1 



Life Insurance. 69 



VALUATION TABLES. 

The net values of different kinds of policies on the 31st 
of December, in each policy year, have been calculated and 
arranged in Valuation Tables convenient for use. With- 
out the aid of these " Valuation Tables," the work of com- 
puting the net value of every policy in all the Companies 
would be an almost impracticable labor. Even with the 
aid of " Valuation Tables," the work is enormous, as may 
be readily comprehended from the fact that one single 
Company has more than sixty thousand policies in force. 

" The Problem that Stands at the Threshold of Life In- 
surance," AND Something that is Within the Threshold. 

It has been previously stated, that, by the payment at any 
age X of the net single premium due to that age, the insured 
places in the hands of a Life Insurance Company enough 
money to pay the cost of insurance during the first policy 
year, and leave in the hands of the Company on deposit, 
at the end of that year, an amount equal to the net single 
premium at the age x-\-l. To effect this, the Company must 
realize net or table interest upon the net single premium, 
paid at the age x. And in like manner the net single pre- 
mium, on deposit at the age x-\-l, will pay for the cost 
of insurance during the second policy year, and leave in 
deposit with the Company, at the end of the second policy 
year, an amount equal to the net single premium at the 
age x-\-2, and so on to the end. 

By the payment, at any age x, of the net annual pi^emium 
due to that age, the insured places in the hands of the Com- 
pany money enough to pay the cost of insurance during the 
first policy year, and leave in the hands of the Company on 
deposit, at the end of that year, an amount sufficient to 
purchase, at that time, a life series of annual premiums, 
each of which is equal to the difference between the net 
annual premium due to the age x, and that due to the age 
a: -1-1. And in like manner, for the following years, to the 
end. 



no Kotes on 

Each of these methods requires a deposit at the end of 
each policy year; and it has been seen that this deposit 
increases every year for each outstanding policy. And in 
time this deposit, in Companies that have a large number 
of policies that have been in force for years, amounts to 
enormous sums of money. The safe and advantageous in- 
vestment of this Trust Fund Deposit is the great " question 
of finance ^^ in Life Insurance. " The proper handling and 
control of these enormous sums of money, belonging to other peo- 
ple, is the onerous trust imposed upon the officers and directors 
of Life Insurance Companies.'''' 

The term " Cost of Insurance,^'' used in this connection, is 
not the actual value of the risk during any year upon an 
amount equal to the amount insured ; because, from the 
moment the insured pays his first premium, the Company 
has in its hands, of the policy-holders' money, enough to 
pay the cost of insurance during the year, and leave the 
requisite deposit at the end of the year. Therefore, from 
the time the first payment is made, the actual risk incurred 
by the Company for the year is not upon the amount of 
the policy, but it is this amount, less the deposit at the end of 
the year. It has been previously seen that we first calcu- 
late the value of the risk on the policy (of ^1), and then 
calculate the cost of insurance (in these two cases of pay- 
ments by net single premium or by net annual premium), 
by finding the amount at risk during the year, and then 
using the following proportion, viz : " The amount of the 
policy is to the amount at risk, as the value of the risk 
during the year on an amount equal to the policy, is to 
the value of the risk on the amount actually at risk." 
The latter is what is generally called " Cost of Insurance." 

It has been seen that the net value that will, at age a:, 

insure one dollar for the first year, is represented by v -p, 

''X 

and that at the same age x, the net value at that time that 
will insure one dollar, to be paid at the end of the second 
year, provided the insured dies during the second year, is 

represented by t^^-^. Similar expressions will give the 



Life Insurance, 71 

means for determining at the age x, what it is then worth 
to insure one dollar, to be paid to the heirs of the insured 
at the end of the n*^ year, provided the insured dies during 
that year. The method of calculating the arithmetical 
value of r, at any given rate of interest, has been already 
very fully explained. Of course, when v is determined, the 
second power, third power, or the n*^ power of ^, is easily 
calculated ; and the Mortality Table gives the number of 
deaths during any year, and the number living at the age 
X. It is, therefore, seen that it is not only very easy {after 
a Mortality Table is furnished^ and a rate of net intei^est is 
fixed upon) to calculate what it will cost, at any age x, to 
insure one dollar, to be paid to the heirs of the insured at 
the end of the first year, in case he dies during the year ; 
but it is easy to calculate what it will cost at age x to 
insure one dollar, to be paid at the end of n years, provided 
the insured dies during the n*^ year. This subject is treated 
at some length in the beginning of these Notes, and an 
arithmetical illustration is given on page 19. The subject 
has been again referred to here, in order to remind the 
reader that " the rate of premium that must be charged, in order 
to carry out an insurance contract, is the problem which stands 
at the threshold of Life Assurance ;''"' and in addition, to call 
his attention once more to that peculiar element in Life 
Insurance which is styled " Reserve " by most writers on 
this subject, and in these Notes is called Trust Fund De- 
posit, and convince him that, after passing the threshold, 
there are some peculiar things connected with this busi- 
ness, as at present conducted, that demand special and 
close examination. 

There is a direct and simple method of insurance (indi- 
cated by the expression v --, as seen above), by which 

the Trust Fund Deposit is entirely avoided; the com^ 
plicated accounts inherent in the ^^ System of Deposits^'' are 
done away wdth ; and Life Insurance, year by year, made 
simpler than by the payment of net annual premiums, 
or net single premiums. But there are serious draw- 
backs in this case, provided a man desires to keep his 



7^ mtes on 

life always insured to the table limit of ninety-nine years. 
But it is not so clear that those persons who desire to in- 
sure their lives, year by year, or for a short period of years, 

would not find it advantageous to pay the amount v -^ 

''X 

each year : or if he desired to insure for ^yq years, and 
avoid medical examinations after the first examination for 
admission, let him pay in advance for the whole five 

years; for the first yediV v —', for the second year v"^ -~^'j 

'-x ^x 

for the third year v^ ~^\ for the fourth year v^-j^; and 

''x ^x 

for the fifth year v'-^. 

''x 

The sum of these yearly amounts, in advance, will pay for 

an insurance for five years. For a limited number of years 

it might be assumed that a higher rate of interest could be 

realized, say the investments were made in safe seven per 

cent, bonds, or bonds and mortgages on real estate worth 

two or three times the amount of the mortgage. In this 

case of interest at seven per cent., v becomes equal to 100 

divided by 107, or $0.934579. At age thirty, it will cost to 

d 727 

insure one dollar for one year, i; -^=$0.934579X 7^ — 7 = 

fO. 007873. This will pay for the insurance of one dollar 
for one year at the age thirty, assuming the Actuaries' Table 
of Mortality, and interest at the rate of seven per cent. 
Multiply by 1,000, and we have $7.87; this amount will 
pay for $1,000 insurance for one year at age thirty. The 
annual premium at same age, actually charged by first- 
class Life Insurance Companies, is about $23.00 — varying 
slightly in the different Companies. But it must be remem- 
bered that the net annual premium, besides paying for Cost 
of Insurance, must provide for the requisite deposit at the 
end of the year ; and the premium actually charged, must 
always be greater than the net annual premium, by an 
amount sufficient to pay all the expenses connected, di- 
rectly and indirectly, with the conduct of the business, 



Life Insurance. 73 

other than the cost of insurance or losses by death prop- 
erly chargeable to this policy. 

Note. — The manner of constructing the Tables, and the arithmetical rules for 
working out the formulas, and the Tables used in making the net calculations, 
have, for convenient reference, been taken out of Part First and placed at the 
end of the book. 



Mr. Gladstone, the Chancellor of the Exchequer, in a speech delivered 
March 7th, 1864, in the British House op Commons, said: 

'•Consider for a moment the peculiar nature of Life Assurance. This is a 
business that presents the direct converse of ordinary commercial business. 
Ordinary commercial business, if legitimate, begins with a considerable invest- 
ment of capital, and the profits follow, perhaps, at a considerable distance. 
But here, on the contrary, you begin with receiving largely, and your liabili- 
ties are postponed to a distant date. Now I dare say there are not many 
members of this House who know to what an extraordinary extent this is 
true, and, therefore, to what an extraordinary extent the public are dependent 
on the prudence, the high honor, and the character of those concerned in the 
manaerement of these institutions. When an institution of this kind is founded, 
so far from having difficulties at the outstart, that is the time of its glory and 
enjoyment. The money comes rolling in, and the claims are at a distance 
almost beyond the horizon." 



Extract from the report of a Committee of the British Parliament in 1853. 
[The party under examination was the Actuary of the "Royal Exchange 
Assurance Office."] 

Question: "Do you think there is any thing peculiar in the character of 
Life Assurance business which would justify the Legislature in interfering with 
it in a way different from other business?" 

Answer: "Yes, both on account of the long period over which the contracts 
extend, and especially for this reason: that Life Assurance offices are now 
taking to make up their accounts on principles that would be scouted from any 
Other department of commercial enterprise." 

Question: "Will you explain what principle you mean?" 

Answer: "The practice of anticipating future profits and treating them as 
assets. Allow me to suppose the case of a bank making up its accounts : it 
owes to its depositors £l*,000,000; it has on hand £900,000; it puts down as 
an additional item of assets, profits, we will say at the rate of £10,000 a year, 
valued at twenty years' purchase; by that means it makes its assets £1,100,000, 
against £1,000,000, and the result is stated to be a surplus of £100,000. That 
principle would never be adopted in a bank, and I think it ought not to be 
adopted in an Assurance Company." 

Question: "But, does it exist in Assurance Companies?" 

Ansiver: "It is done." 

Question: "Is it done by Assurance Companies generally, or only in particu- 
lar cases? " 

Answer: "It is in considerable use, and the practice is extending." 



"There can hardly be a happier set of capitalists on earth than one which 
has obtained a right, by perpetual charter, to insure lives, receiving from the 
proceeds, first legal interest semi-annually on stock as a sure thing, and sec- 
ondly, twenty per cent, of what are called profits, that is premiums proving 
to be surplus, if it can once secure a flourishing business." 

Elizur Wright (1862). 



PART SECOND 



PRACTICAL LIFE INSURANCE. 

Expenses — Loading — Surplus. 

In order to pay the Expenses of conducting the business 
of " Practical Life Insurance," it is necessary that addi- 
tional means should be provided over and above the net 
annual premiums; the latter being enough, and only enough, 
when regularly compounded yearly at four per cent., to pay 
the Cost of Insurance, and furnish the requisite Deposit or 
" Reserved 

It is usual to add to the net annual premium from tvi^enty 
to thirty per cent., or even more, for the purpose of defray- 
ing expenses. This addition to the net annual premium is 
technically called Loading. 

The Loading may, and often does, more than pay ex- 
penses. 

The interest actually received is nearly always more than 
the net interest assumed in the table calculations. 

And the actual mortality, particularly in the earlier years 
of a Company, is, in practice, generally less than that given 
in the table. 

From each of the three above-named sources Surplus may 
be obtained. By Surplus is here meant money^ or its equiv- 
alent, in excess of what is required to pay losses by death 
during the year, to form the " Deposit " for the policy at 
the end of the year, and pay all expenses. The surplus, 
in purely Mutual Companies, belongs to the policy-holders. 
In the purely Stock Companies all the surplus goes to the 
share-holders. The mixed Companies are those Stock Com- 
panies that give some portion of the surplus to the policy- 
holders. 

In order to investigate the nature of practical Life Insur- 
ance business for one year, let us suppose that the Cost of 



76 Jfotes on 

Insurance, and all expenses of the previous year, have been 
paid, and that the Company had on hand, at the close of the 
previous year, the requisite Deposit for each and all of its 
outstanding policies. We will, for the present, suppose that 
the Surplus of the previous year had been distributed to its 
respective owners. 

At the beginning of the year, the business of which, we 
are now investigating, each policy-holder pays his full an- 
nual premium ; that is, the Loading is included. There is 
then in the hands of the Company, on account and to the 
credit of each policy, the two amounts, viz : the Deposit at 
the end of the preceding year, and the full annual premium. 

These sums are both invested at the best, safe rate of 
interest ; and out of these two amounts, thus increased by 
interest, actually received during the year, the " expenses" 
for the year, properly chargeable to each policy, must be 
paid. The Cost of Insurance, or proportion of losses by 
death during the year, properly chargeable to each policy, 
must be paid. And the requisite Deposit at the end of the 
year for each policy, must be provided and set apart, or 
securely invested for the policy-holder at the net, or table 
rate of interest at least. If there is anything left on account 
of each policy, it is Surplus produced by the policy. 

(1.) Let (T. F. D.)^^„ (1 + ?^') represent the Deposit at the 
end of the n^^^ year from the date of the policy^ increased by the 
interest actually received upon it during the year now in ques- 
tion ; i. e. (n^^~{-l). r represents the rate of interest divided 
BY 100. 

(2.) Let (P^ — e) (1-l-r') represent that the expenses chargeable 
to this policy, for the year, are taken out of the full annual pre- 
mium, and the remainder of the premium is increased by the 
interest actually received upon it during the year. In this, P 
is the gross or full premium ; e represents expenses ; and r , as 
above, represents the rate of interest divided by 100. 

(3.) Let .y^Yl— (T. F. D.)^_^„_j.i) represent the actual " Cost 

OF Insurance " during the year ; d' represents the actual num- 
ber of deaths during the year ; I' represents the number actually 
living at the beginning of the year. 



Life Insurance. 7 7 

(4.) Lei (T. F. D.)^^„4.i represent the Deposit at the. end 
of the 'n}^-\-l year of the policy. 

Add expression (2) to expression (1); subtract expressions 
(3) and (4) from the sum thus produced. If there is any- 
thing left, it is Surplus produced by this policy. 

Combining the expressions (1), (2), (3), and (4), as above 
indicated, we have the equation — 

(T. F. D.U„(l+'-')-f(^-^)(l+'-')-^"(l-(T. F. D) 

a;-j.„+l )— (T. F. D.)^_|.^_j_i=SuRPLUS PRODUCED BY THIS POLICY DUR- 
ING THE n*''-|-l YEAR. 

When the Surplus arising from the funds of each policy 
is obtained as above, and is distributed in accordance with 
the principle expressed in this equation, it is said to be 
divided upon the " Contribution Plan." 

Explanation of the Terms Contained in the Above EauA- 

tion. 

Notwithstanding the rather formidable appearance of the 
signs and symbols in the above equation, it is really very 
simple and plain, when expressed in the ordinary words of 
our language. 

The first term expresses only that the Deposit on hand, at 

the end of the n^^ year, has been increased by the amount 

it actually produced when placed at the best safe rate of 

interest during the ?z*''+l year, r is not the rate of interest, 

but is a quantity which, when multiplied by the Deposit, 

will give us the amount actually received from interest 

during the year. Suppose this rate of interest had been 

7 
seven per cent., then r is equal to . The second term 

expresses that from P, the full annual premium, we sub- 
tract the expenses actually incurred and properly chargeable 
to this policy during the ?i'''+l year, and place the remainder, 
which is (P — e), at the best safe rate of interest; and 
add the interest so obtained, to what was left of the full 
annual premium after the expenses had been deducted. 



78 Jfotes on 

The third term is the actual Cost of Insurance during the 
n^^J^l year, properly chargeable to this policy. In which 
/'^^.^ represents the number actually living at the end of 
n years from the date of the policy ; that is, at the begin- 
ning of the n^^'\-\ year; and d'^^^ represents the number that 
actually die during the year between age x-\-n and the 
age a^-j-w+l- The second factor of this third term is the 
amount the Company has at risk during the year. 

The fourth term is the requisite Deposit that must be on 
hand at the end of the n^^Ar^ year from the date of the policy. 

Variations from the Table Rate of Mortality. 

We have already explained how to calculate the requisite 
Deposit. The Amount at Risk, too, is a specific sum, and 
is equal to the amount of the policy, minus the Deposit for 
the end of the year in question. But the fraction obtained 
by dividing the number of actual deaths during the year 
by the actual number living at the beginning of the year, 
occasionally varies somewhat, and may vary greatly, from 
the general law of mortality amongst large numbers of 
mankind. This is particularly liable to be the case when 
the Company has but a limited number of policy-holders ; 
and the variation would be very marked, whether the whole 
number of policy - holders in the Company was large or 
small, in case sickly or diseased persons were taken by 
the Company. It is worthy of notice, too, that if the num- 
ber of deaths, in any one year, should prove to be remark- 
ably small, it is not safe to assume that, because the losses 
by death, in that year, are greatly less than those called for 
by the Table of Mortality, that the difference is clear gain, 
and can be disposed of as " Surplus," and distributed at the 
end of the year; because the variation from the number of 
deaths called for by the Table of Mortality, will probably 
soon vary on the other side ; and, when the losses are ma- 
terially greater, by death during the year, than called for by 
the table, these losses have to be paid, and that promptly. 
It is, therefore, well to have something in hand to pay with. 
The Company cannot touch its Deposit, because that money 
does not belong to the Company. It must not only be held for 



I 



Life Insurance. 79 

its proper owner, but it must always be earning interest for 
its owner ; and this interest must be regularly compounded 
every year. Purely Mutual Companies, if well conducted, 
always keep back a portion of their Surplus. One of these 
Companies, in its last official report, has kept back, and no 
doubt wisely, nearly two and one quarter millions of its 
Surplus. It is a large Company, has large risks, and ought 
to endeavor by all prudent means to meet its risks. 

Effect of Allowing a Policy for a Large Amount to be 
Grouped With Those of Small Amount. 

Again referring to the term in the above equation that 
expresses the Cost of Insurance, properly chargeable to 
the policy for the year, it will be noticed, that although the 
Amount at Risk on the policy is a factor in that term, and 
to this extent, the amount of the policy, however large, has 
been fully considered and allowed for, still, if a one hun- 
dred thousand dollar policy had been, by the Company, 
grouped in the accounts with a number of policies of 
smaller amount, but of the same age, say with ninety-nine 
others of one thousand dollars each, the death of the single 
individual in this case would be a greater loss to the Com- 
pany than that of the whole other ninety-nine policy-hold- 
ers. These things, and many others of a similar nature, 
have to be closely watched and strictly attended to by 
officers of the greatest skill and good judgment. And 
whilst merely theoretical information is not enough to qual- 
ify an officer to conduct this or any other business success- 
fully, it may be set down as certain that no man can conduct 
Life Insurance safely and properly unless he knows some- 
thing of the principles upon which it is founded. 

Further Allusion to Loading and Expenses. 

Let us now consider the Loading which has been added 
to the net annual premium, and the Expense which this 
Loading is intended to provide for. In the first place, it 
may be remarked that the " Expenses," properly chargeable 
to a policy, are not necessarily the same proportion of the 
annual premium in different cases. At the end of the year, 



80 J^otes on 

although it may require some labor to adjust with precision 
the Expense account for each separate policy, or each dis- 
tinctive set of policies, this should be attempted, and sub- 
stantial equity in this respect can always be attained. 

If the amount of Expenses, and the interest that will be 
realized in the most unfavorable year, during the contin- 
uance of a policy, could be accurately ascertained in ad- 
vance, the Loading would, even in that case, have to be 
enough to meet the worst 3-ear. 

Certainty of the payment of his policy at maturity is 

WHAT EVERY MAN WANTS V/HEN HE INSURES HIS LIFE. TllC qual- 
ity of the article he purchases is the first and the greatest 
consideration with him. It must be certain that the Life 
Insurance Company has charged enough to enable it to 
pass safely through the worst years that can reasonably be 
expected to occur during the period of the contract between 
the Company and the insured, which is generally supposed 
to be for a lifetime. 



Life Insurance, 81 



SURPLUS. 

In favorable, or even in ordinary years, the Loading, 
and the interest on the funds of the Company (because of 
their realizing usually a higher rate than that called for by 
the table calculations), v^ill produce a "Surplus" on each 
policy at the end of the business of a year. This Surplus 
arises from previous over-payment in advance, demanded by 
the Company, in order to make it certain that the Com- 
pany w^ill be self-sustaining in the w^orst year that may 
occur in a lifetime. The Surplus distributed to policy- 
holders is merely a return to them of that part of the 
premium they paid at the beginning of the year, w^hich, 
at the end of the year, is found not to have been required 
during the year, either in effecting the insurance, providing 
the means (the Deposit) for paying the policy at maturity, 
or in paying " expenses." 

Accounts to be Kept with Each Policy. 

It must always be held in mind, that whilst in Life Insur- 
ance there are peculiar and mandatory arithmetical laws by 
which particular money values are computed — in addition, 
and after these values are accurately determined — practical 
Life Insurance becomes like all other business which in- 
volves the handling and control of vast amounts of money. 
Good judgment, great industry, the strictest integrity, and 
sound practical business sense on the part of those intrusted 
with the conduct of Life Insurance, are all absolutely essen- 
tial to successful management. 

No prudent man will ever attempt to control or conduct 
any important business without making some kind of esti 
mate in advance. The Mortality Table furnishes the means 
for making certain estimates with an accuracy that is not 
usually found in ordinary business. But, in regard to the 
"Expenses" that will be incurred, or the rate of interest 
that will be realized on these investments, or the bad in- 
vestments that may be made, or whether some of its oiH- 
cers may not prove to be dishonest, or what the Company 
will make or lose, and a variety of highly important ques- 
6 



82 mtes on 

tions of this nature, cannot be settled by estimates made 
beforehand by Life Insurance Companies any more defin- 
itely than similar estimates can be made in any other busi- 
ness. Nevertheless, these estimates of practical results 
ought always to be made in advance ; but it will be a 
sad mistake to assume, at the end of a year, that the esti- 
mates made at the beginning of the year give, necessarily, 
an accurate statement of the real condition of the business. 
At the end of the year the estimates must be changed to 
fit the facts. 

The question of Expenses in Life Insurance, as in every 
other business, is a matter of vital importance, and ought 
to receive the closest attention, not only from those who 
manage the business, but from those who keep the accounts. 
It is not proposed here to intimate even how the accounts 
of a Life Insurance Company ought to be kept; but only 
to allude to some of the main practical, ordinary matters, 
that are just as essential in Life Insurance as they are in 
all other kinds of business. When at the end of a year 
the Expense account is made up, and the amount prop- 
erly chargeable to each policy has been determined, e, in 
the equation, becomes known ; r' is the one hundredth 
part of the average rate of interest actually received by 
the Company during the year upon its aggregate invest- 
ments. The manner of calculating the net annual premi- 
um has already been explained. Add to this the Loading 
that has been fixed upon, and P, in the preceding equation, 
becomes known. The manner of calculating the Deposit 
at both the beginning and the end of the year, and the Cost 
of Insurance during the year, has been previously given. 
There is nothing further needed, in order to make up the 
accounts of each separate policy for the year, after the gen- 
eral accounts of the year's business are made out. It must 
be noticed, however, that in case there are any items in the 
general accounts, involving either gain or loss, that do not 
enter, in some shape, into the separate policy accounts, the 
aggregate surplus, arising from the separate policies, will not 
be equal to the surplus shown by the general accounts to be 
actually on hand. It no doubt happens often that certain 
gains or losses during the year are not carried into the sepa- 



1 



Life Insurance. 83 

rate policy accounts on the books of a Company. In this 
case, when the amount of surplus produced by a policy is 
calculated by the above equation, the actual surplus for the 
policy will be obtained by the following proportion, viz : 
the total surplus obtained by adding together that of all 
the separate policies is to the actual surplus, as shown by 
the general accounts, as the surplus for each particular 
policy is to the amount of surplus to be credited to the 
policy in the general accounts. 

In case this account includes everything not included 
in the other terms of the equation for obtaining the sur- 
plus, there will be no occasion for using the proportion 
just referred to ; because, in this case, the sum obtained 
by adding together the surplus of all the separate policies 
will be just equal to the surplus shown by the general 
accounts. 

The subject of accounts in Life Insurance Companies will 
never be definitely settled until the book-keepers and ac- 
countants clearly understand the theory and principles upon 
which Life Insurance is founded. It is safe to say, that if 
any money account is kept with a policy at all, it ought to 
be made exactly correct. 

Example Illustrating the Calculation of Surplus. 

The following arithmetical example is given merely in 
illustration of the method of making up the accounts of a 
policy for any year, and determining the surplus : 

It is assumed that an ordinary whole life policy for one 
thousand dollars, taken out at age forty-two, is in its 10th 
year. The net annual premium, as previously calculated, 
is $25.55 ; take the loading to be 33J per cent, of this ; then 
the full annual premium is $34.05. To make out the ac- 
count of this policy during the 10th year, we w411 assume 
that the expenses properly chargeable to it during the year, 
are, at the end of the year, found to be twenty per cent, of 
the full or gross annual premium; that everything to the 
credit of this policy at the end of the preceding year, ex- 
cept the Deposit, had been distributed to its owner or 



84 J^otes on 

owners; that the rate of interest actually realized by the 
Company on its aggregate investments during the year, was 
seven per cent.; and that the mortality amongst the insured 
during the year, was that call-ed for by the table. 

The first thing to be done, is to calculate the Deposit for 
the end of the preceding year : 

.T=42, 71=10. Then (T. F. D.), . ._i, or (T. F. D.)5i=l— ^. 

j3 ^ . N51 121983.7 __ i^Q,on«. 

But A5i= =^= = 18.179206; 

'' D^i 9255.77 

and A^= ^ = ^^^^^^ ^ ^15.621223; 
Di2 l4&au.55^ 

therefore ^^ J-It^^^ $0.843673; 
A42 15.621223 

A 

and l—-i'=$l—$0.843673=$0. 156326; 

A42 

which is the Deposit for an ordinary whole life policy for 
one dollar at the end of the 9th year, the policy having been 
taken out- at age forty-two. In a precisely similar manner, 
we calculate the Deposit for this policy at the end of the 
10th year, only noting that, in this case, the formula is 

(T, F. D.),+„, or (T. F. D.)52=l— ^'. The Deposit for the 

end of the 10th year will be found to be $0.175155. 

The Amount at Risk during the tenth year on this policy 
of one dollar, was $1 — $0.175155 = $0.824844. The mor- 
tality amongst the insured during the year having been 
assumed to have conformed to that expressed in the table, 

we find from the table that — ''- is the fraction that 

68,409 

expresses the rate at which the insured persons died during 
the year. This fraction, multiplied by the Amount at Risk 
during the year, gives $0.01393 as the amount properly 
chargeable to this policy of one dollar, during the year, for 
Cost of Insurance. 

Multiply each of the above Deposits, and the Cost of In- 
surance by 1,000, and we have the Deposit at the end of 
the ninth year equal to $156.33; the Deposit at the end of 



Life Insurance, 85 

the tenth year equal to $175.16; and the Cost of Insurance 
during the tenth year equal to $13.93 on an ordinary whole 
life policy for $1,000 taken out at age forty-two. 

We are now Ready to Make out the Account of this 
Policy for the Tenth Policy Year. 

The Deposit at the end of the ninth or beginning of the 
tenth year was $156.33. The full or gross annual premium 
is $34.05, twenty per cent, of which is deducted for ex- 
penses ; leaving on hand at the beginning of the year 
$27.24 of this premium. Add this to the Deposit on hand 
at the end of the preceding, or beginning of the present 
year, and we have to the credit of the policy $183.57, after 
expenses for the year have been provided for. 

Add to this, interest for one year, at seven per cent., 
which is $12.85, and we have to the credit of the policy, 
in the account for the tenth year, $196.42. Deduct from 
this the Cost of Insurance, $13.93, and we have $182.49. 
Take out the Deposit that the Company must have on hand 
for this policy at the end of the tenth year, $175.16, and 
there is $7.33 Surplus, which is about twenty-one per cent, of 
the full annual premium paid. 

But in case the expenses for the year, and the mortality 
amongst the insured, had been greater than that assumed 
in this example, and the interest had been less, this surplus 
would have been diminished. On the other hand, had the 
variation in expenses, mortality, and interest, been the op- 
posite of the above, the Surplus would have been greater. 
Certainty of payment of his policy at maturity is what every 
policy-holder wants ; and it is fair to suppose that both he 
and his heirs would prefer to have the whole amount paid 
in money. We have seen above, that, with a Loading of 
33^ per cent, on the net annual premium, there was, at the 
end of the year, a Surplus of $7.33 : no great margin, when 
the question is that of the prompt and certain payment at matu- 
rity of a policy of one thousand dollars, 7?iore especially irhcn 
the Surplus or over-payment made at the beginning of the year, 
IN order to make the payment of the policy safe, is returned 
TO the policy-holder at the end of the year. 



86 J^otes on 

When the Surplus belonging to the policy-holder is not 
distributed, but remains in the hands of the Company to 
the credit of the policy that produced it, it ought to be in 
vested for the holder of the policy. When the Surplus has 
all been distributed, the true value of the policy at the end 
of any year, and before the payment of the next annual 
premium, is the Deposit; but when it has not been distrib- 
uted, the true value of the policy is the Deposit, plus any 
Surplus there may be in the hands of the Company to the credit 
of the policy. 



1 



Life Insurance, 87 



ADDITIONAL INSURANCE PURCHASED WITH THE SURPLUS, 

When the Surplus is distributed to the policy-holders, it 
may be used in part payment of the next annual premium, 
or, at the option of the policy-holder, it may be applied to 
the purchase of additional full-paid insurance. The latter 
would progressively increase the amount of the policy ; the 
former would result in a progressively diminishing annual 
premium. 

When the amount of Surplus to be returned has been 
determined, the amount of full-paid insurance that this Sur- 
plus will purchase at that age is calculated by first find- 
ing the net single premium that will insure one dollar at 
that age. This we have agreed to represent by ^P^ ; and, 
as previously shown, sV^=\ — (1 — v) K^, and its value is 
calculated by replacing x by the age in question ; and ob- 
taining the values of N^ and D^ opposite that age in the 
table; this numerator and denominator gives us a fraction 
which is the value of A at that age. The numerical value 
of V has previously been determined; and we can, therefore, 
convert the second member of the above equation into its 
proper arithmetical value. This will give the net single 

PREMIUM sP^, WHICH, AT THAT AGE, WILL INSURE ONE DOLLAR. Of 

course the surplus, used as a net single premium at that 
age to purchase additional full-paid insurance, will buy a 
proportional amount. 

Take, for example, age thirty. Suppose that the surplus 
is $15.36, and the policy-holder desires to purchase with this 
an addition to his policy, instead of using it in part payment 
of his next annual premium. 

First find what net single premium will, at that age, in- 

N. 
sure one dollar. The equation becomes 5Pjo=l — (1 — '^)t^^' 

From the table we find N3o=479.95l.6 ; and 0,0=26.605.37. 
As before calculated, we have ?;=$0.961538. Therefore 
(1— 'y)=$0.038462. Multiply this by N,,, and divide the pro- 
duct by D30. Subtract the result from unity, and we have 
$0.306158. This is the net single premium that will, at 
age thirty, insure one dollar, to be paid to the heirs of the 



88 J^otes on 

insured at the end of the year in which he may die. And 
the question in arithmetic is this: if ^0. 306158 will insure 
one dollar, how much will $15.36 insure? The answer is, 
^50.17; and this is the amount of addition to the policy 
that the surplus named will purchase. This additional in- 
surance is full paid, and the $50.17, in this case, is called by 
insurance writers the " reversionary value " of the surplus, 
$15.36. 

Any proceeds that may in the future arise from interest 
on this $15.36, in excess of the four per cent, necessary to 
pay the Cost of Insurance and provide the requisite Deposit, 
will be additional surplus, and may be used as it accrues in 
purchasing additional full-paid insurance. It is estimated 
that a whole life policy for $10,000, taken out at age thirty, 
by the payment of an annual premium of $230.20 every 
year for twenty-three years, will, in that time, amount to 
$17,906.50, if the surplus arising from over-payments is ap- 
plied to the purchase of additional full-paid insurance; and 
the surplus arising from interest over four per cent, upon the 
net single premium, paid for the additional insurance, is 
applied yearly to increase the policy. Moreover, at the 
end of twenty-three years, the policy will have become vir- 
tually full paid ; because it is estimated that the return of 
surplus, after that time, will exceed the annual premium, 
and there will be no further payments required. On the 
contrary, as the return of surplus now exceeds the annual 
premium, the policy-holder may receive an income, or real 
dividend, from his policy; or by leaving the money in the 
hands of the Company, his policy will go on increasing, 
from year to year, without any further payment of pre- 
miums until it matures at his death. 

The peculiar and attractive feature in cash Companies 
that return the surplus to the owner of the policy, either 
in cash, diminishing by that amount the annual premium 
required, or by purchasing with the surplus additional full- 
paid insurance, is virtually unknown in note Companies or 
in the strictly proprietary Companies. In the latter all the 
surplus goes direct to the share-holders; and the insured 
pays a given amount every year, without addition or diminu- 
tion of his annual premiums or of his policy. In the note 



Life Insurance. 89 

Companies the surplus to policy-holders generally consists, 
in great part, of their own notes; and there is, in most 
cases, no diminution of the amount of premium required 
from year to year; but there is often a diminution in the 
amount of the policy, arising from an accumulation of notes 
of the policy-holder, which must be deducted from the policy 
at maturity. It is believed that policy-holders who have 
been insured in note Companies for a period of years, are 
quite sure to realize the fact that the burden upon them is 
getting heavier instead of lighter, and that it is only those 

WHO DIE SOON after GETTING INSURED THAT CAN POSSIBLY GALN 
ANY ADVANTAGE FROM THE SYSTEM OF INSURING UPON CREDIT, 
RATHER THAN FOR CASH. 



90 Jfotes on 



Comparison Between a Cash Company that Returns the Surplus 
to the Policy-holders and a Note Company that Retains Twenty 
Per Cent, of the Surplus. 

In illustration of some of the features of the system of 
Life Insurance on credit, and the effect of appropriating to 
the shareholders twenty per cent, of the surplus of the 
policy-holders, we will, for illustration, suppose a Com- 
pany has $100,000 full-paid capital stock. It receives one- 
third of its premiums in notes, or makes a loan of that 
amount to the policy-holders; and by its charter has se- 
cured the right to twenty per cent, of the surplus in cash, 
whilst eighty per cent, of the surplus is returned to the 
policy-holder by canceling on the books that much of a 
LOAN the Company is supposed to have made to him. The 
policy is for whole life $10,000, and taken out at age 
thirty. The annual premium in this Company is $233.00. 
The annual premium on a similar policy in the All-cash 
Company, with which the loan is proposed to be compared, 
is $230.20. 

At first sight it might appear that the Loan Company has 
the advantage in price, because its premium is only $2.80 
more than that in the All-cash Company, and it loans the 
policy-holder one-third of the money, and only charges him 
six per cent, interest on the loan. But let us take up the 
accounts of this policy in the Loan Company, and see how 
it will stand at the end of the first year, and then take it 
year by year to the limit of the table ; that is, up to, and 
including, ninet3^-nine years of age. 

By the formula previously deduced, let us calculate the 
Deposit that must be on hand at the end of the first year 
in order to make the payment of the policy in money secure. 
It will be found to be $93.10. 

By the Table of Mortality we use, which is the Actuaries', 
and four per cent, net interest, the cost of insurance for the 
year will be found to be $83.46. But it is claimed by some 
Companies that "American Experience" among insured lives 
is more favorable than the above ; and that by this table, 
with net interest at four and a half per cent., twenty per 
cent, of this Cost of Insurance may be abated. We will, 



Life Insurance. 91 

for the purposes of the present examination and compari- 
son, assume that the Cost of Insurance, in practice, is only 
four fifths of what is called for by the calculations, based 
upon the Actuaries' table, four per cent. We therefore take 
this Cost of Insurance as being $66.77, instead of $83.46. 

It is assumed that the expenses will be twenty per cent, 
of the premium, or $46.60 per year. On account of this 
policy, then, for the first year, there must be paid for Cost 
of Insurance, $66.77; for expenses, $46,60; and a Deposit 
of $93.10 must be on hand at the end of the year. This 
makes the liabilities of the Company on this policy, for the 
first year, amount to $206.47. 

Let us now see how the other side of the account of this 
policy stands. The annual premium is $233.00; but $77.67 
of this is LOAN. Cash, $155.33. Six per cent, cash in ad- 
vance on the loan is $4.66. Add this to the cash part of 
the premium, and we have $160.00 cash received by the 
Company at the beginning of the year. Place this at seven 
per cent., which is the rate we have assumed the Company 
will realize, and regularly compound during the existence 
of the contract, or for sixty-nine years. 

The interest upon $160.00 for one year would, at seven 
per cent., be $31.20; hence, the principal and interest 
amounts to $171.20, with which to meet the liabilities for 
the year, viz : $206.47. This shows a deficiency of cash at 
the end of the first year amounting to $35.37. But by 
counting the loan as an asset, and adding it to the cash, 
we can make out a Surplus, and make so-called dividends to 
policy-holders, and real dividends to share-holders. The 
loan is $77.67; add this to the cash, $171.20, and we have 
$248.87 with which to meet $206.47 of liabilities. This 
makes the Surplus $42.40, twenty per cent, of which, or 
$8.48 IN cash, goes to the share-holders; but as there is no 
cash in this so-called Surplus, the cash to pay the share-hold- 
ers MUST be OBTAINED FROM SOME OTHER SOURCE. Wc haVC SeCH 

that the calculations call for $03.10 in Deposit; but after 
paying the Cost of Insurance and expenses, there was left 
only $57.83 in cash. That the share-holders may receive 
their twenty per cent, in cash, the amount $8.48 must be 
taken from the already inadequate Deposit, and its place 



92 JVotes on 

there be supplied by an equal amount taken from the Sur- 
plus, all of which is loan. The dividend to the policy-holder 
is made by canceling on the books an amount equal to eighty per 
cent, of the Surplus loan, viz : $33.92, whilst the share-holder 

TAKES INTO HIS PRIVATE POSSESSION AND OWNERSHIP $8.48 IN CASH ; 

and the Comjjany has in Deposit, at the end of the first year, but 
$49.35 in money, instead of the $93.10 it should have in order 
to secure the payment of the policy in money at maturity. 

At the beginning of the second year there is in the hands 
of the Company, to the credit of this policy, this fragment 
of the- Deposit that the calculations call for on a cash basis. 
The annual premium is paid as before. Deduct twenty per 
cent, of the premium for expenses; place the money that 
is left in the hands of the Company at seven per cent.; cal- 
culate the Deposit for the end of the second year ; calcu- 
late the Cost of Insurance during the jear; find the Surplus- 
as before ; take twenty per cent, of the Surplus, and place 
it in the Deposit ; take an equal amount of cash out of the 
Deposit, hand it to the share-holders, and give eio-hty per 
cent, of the Surplus, w^hich is all loan, to the policy-holder, 
for his DIVIDEND on the business of the second year. 

The calculations have been made following the accounts 
of this Company, year by year, to, and including, age ninety- 
nine. They are not given here, because it is believed that 
the tables and formula will enable any tolerably fair arith- 
metical computer to make them for himself. It is proposed 
here to give only a brief summary of the main facts devel- 
oped by the calculations. In the first place, it was fifty 

YEARS BEFORE THE INSURED COULD WORK HIS POLICY OUT OF DEBT. 

At the age of eighty he was for the first time really insured 
for $10,000, and he had by this time paid exactly $8,000 m 
cash to the Company, and he will have to pay $160.00 cash 
the next year. He has now^ reached, however, the point at 
which his annual premiums may be reduced by a return of 
real Surplus ; or, if he prefers to continue to pay the $160.00 
annually, his policy may now begin to increase. 

Let us look at the policy in the All-cash Company. We 
will not refer to the detail calculations of the first year, 
because there are no complications in this case — it is all 
CASH ; no cutting up of amounts and taking notes or loans 



Life Insurance. 95 

from one place, to replace cash removed from another. We 
find that in the Cash Company the policy had become vir- 
tually paid up by the end of the thirty-eighth year, because 
the return of annual surplus in cash had by that time be- 
come equal to the annual premium. The insured had paid, 
in all, $4,077.94, and v^as done paying ; but his return of 
surplus will continue to increase ; and so far from having 
any thing to pay on his policy after the thirty-eighth year, 
he will receive an annual cash dividend from it, or income. 
Not one of those '^ so-called^'' dividends, by which I give you 
$100 and my note for $30 now; and at the end of one year 
I give you another $100 in cash and my note for $30, and 
you hand me back the first note for $30 and call that a 
" DIVIDEND " of thirty per cent., and make me believe that 
I have made a thirty per cent, investment,, and am getting 
rich, when, in fact, there has been no real use for the note 
or LOAN, except to return it to me, or give it to my heirs as 
money in case I die. The policy-holder in the Cash Com- 
pany above described is done paying at the end of thirty- 
eight years for the policy he bought, and is now receiving 
a clear income from the surplus price he paid for it. We 
have seen the man in the Loan Company reach the fiftieth 
year. He had paid $8,000 cash, and was still paying $160 
per year in money ; but is now for the first time relieved of 
the necessity for accepting loans from the Company. 

Let us see how the accounts of the old man's policy will 
stand at the end of the sixty-ninth year, on the supposition 
that his surplus had been returned to him in part payment 
of his cash annual premiums since he passed eighty years. 
We find him at the beginning of the sixty-ninth year only 
required to pay, in cash, $79.10. This, and the return sur- 
plus he was entitled to, makes up the cash part of his regular 
annual premium, $155.33. We find his Deposit at the end 
of this year to be $9,445.70. If to this we add the net an- 
nual premium, actually due to the age tliirty, at which lie entered 
the Company, $169.70, instead of the cash part of the annual 
premium he has been paying, we find the sum in the hands 
of the Company, to the credit of this policy-holder, the day 
he is ninety-nine years old, will be $9,615.40; and this, at 



94- ' Jfotes on 

four per cent., will, by the end of the year, produce $384.61 
interest, which, added to the above sum, makes $10,000. 

We accompanied the old man to the limit of the table, 
not that we had any doubt about his heirs getting the 
money, but we desired to see how much cash he w^ould have 
to pay, and how much of his money would go direct into 
the hands or pockets of the share-holders. It appears that 
he paid in cash $9,793.40, of which the share-holders, under 
their charter rights, took $1,652.00 into their own private 
possession. The policy-holder in the All-cash Company, as 
we have seen before, had fully paid up at the end of the 
thirty-eighth year, and had paid in all but $4,077.94, after 
which his policy increased progressively in amount, or, at 
his option, it yielded him a yearly income. We find in the 
whole transaction the share-holders in the Cash Company, 
although handsomely paid for their attention to the business, 
received only $73.76, and this was from interest on the ad- 
ditional guarantee fund, which belongs to the policy-holders 
in that Company — is held for the security and benefit of the 
policy-holders, and yields sufficient interest to pay the share- 
holders a VERY LIBERAL, but uot exorbitant, compensation for 
their personal attention to the business of the Company. 



Lif& Insurance. 95 



COMIVIENTS UPON INSURANCE PARTLY ON CREDIT. 

A Life Insurance Company can, with safety to itself, 
accept the notes of a policy-holder in part payment of the 
"net annual premium," and the amount of these "notes" or 
"loans" may equal, but must not exceed, the Deposit. The 
Deposit increases from year to year, and the notes or loans 
may be increased to the same extent, but no more. The 
notes or loans must be deducted from the face of the policy 
at maturity, therefore, the amount actually insured in money 
becomes less and less each year. The question is not, " can 
A Life Insurance Company safely accept notes in part pay- 
ment OF THE ANNUAL PREMIUMS ?" But rathCT, " CAN A POLICY- 
HOLDER, FOR ANY GREAT LENGTH OF TIME, AFFORD TO ACCEPT THE 
CREDIT PROFFERED BY THE LiFE INSURANCE CoMPANY ? " 

Suppose that we take this case to the limit of the table, 
ninety-nine years. The policy-holder will have paid each 
year, his proportion of the losses by death — called Cost of 
Insurance — and the yearly expenses ; and the Deposit, con- 
sisting entirely of his own notes, will have amounted to 
within a very small fraction of the whole amount of the 
face of his policy. The man dies in the one hundredth 
year of his age, and the heirs receive his notes in part 
payment of the policy ; and these notes are, in this par- 
ticular case, enough to FULLY PAY THE POLICY WHEN THE LAST 
ANNUAL PAYMENT ONLY, IN MONEY, IS ADDED TO THESE NOTES. 

This certainly is not a desirable kind of Life Insurance 
for those who live long. On the other hand, if the insured, 
dies early, he will gain by the note, or loan system; but 
what he gains by notes or loans, the man in that Company 
who lives long loses. 

The Life Insurance Company is safe in this case, provided 
it has a large number of policy-holders, and retains them to 
the end of their lives. 

It is true that Note or Loan Companies seldom, if ever, 
in practice, push the credit system to the extreme limit 
given above; but they may do it with safety under the 
above proviso. The question is, can the policy-holder stand 
it if he does not die soon? 



96 J^otes on 



RETURN PREMIUM PLAN. 

There is a comparatively new, and, at first glance, to some 
persons, attractive kind of whole Life Insurance, by which 
the Company contracts not only to pay the amount of the 
policy at maturity, but also to return, at the death of the 
policy-holder, the full amount of all the premiums he has 
paid. 

This, of course, cannot be done by the Company unless 
the policy-holder pays for all he gets. It appears by the cal- 
culations made in these cases for the return of premiums, 
that it requires $53.65, net annual premium, to insure $1,000 
at age forty-two and return the premium. But $25.55 is the 
net annual premium that will insure $1,000 at the same age 
on the ordinary life policy. Twice this amount, or $51.10, 
will insure $2,000 for the first year, whereas $53.65, on the 
Return Premium Plan, will only insure for the first year 
$1,053.65. It is hardly necessary to continue the discussion 
of the Return Premium Plan through its mathematical com- 
plexities. It makes too bad a commencement at Life Insur- 
ance, and a man would have to be certain of living nearly 
a quarter of a century before he could hope to gain any 
thing by this plan. 



Life Insurance, 97 

GENERAL COMMENTS. 

It is essential to the policy-holder that the Life Insurance 
Company with which he may take out a policy, should be 
controlled by wise and stringent laws, rigidly enforced ; be- 
cause, from the nature of this business, the funds held in 
trust are peculiarly liable to misapplication. Nothing short 
of the searching probe of stringent and wise laws, rigidly 
enforced, can prevent this, in case the managers of such 
Companies are not thoroughly acquainted with the princi- 
ples upon which Life Insurance is founded, and firmly de- 
termined to adhere to them. They must attend closely to 
every detail ; make no mistakes in their risks or invest- 
ments ; and should be men of the greatest integrity of char- 
acter and sternest honesty of purpose. But, to insure safety 
in the business, every detail should be furnished, at least once 
in every year, to some competent State officer; and by the 
latter, the accounts should all be carefully recomputed, and 
the results published. Sound and well-conducted Companies 
desire this, and others should be forced to a full exhibit of 
all their affairs. 

Borrowing Great Names. 

"The device of borrowing great names has, from the com- 
mencement, been resorted to by the projectors and managers 
of many Life Insurance Companies. The facility and read- 
iness with which men of influence lend their names to busi- 
ness, of which they know little and care less, has long been, 
and is yet, a short and easy road to temporary popularity 
and public favor for business, that has no sound and legiti- 
mate claim to the confidence and respect of the community." 
An array of great names in Life Insurance is not enough. 
Neither is it a very sound business principle to trust, in a 
matter of this importance, to the mere request or solicitation 
of an agent. It is not enough that the agent may point out 
in a table of figures the amount a person would be required 
to pay for a certain kind of policy, and in his own mind 
figure up what his (the agent's) commission would amount 
7 



98 Kotes on 

to IN CASH, on the premium he asks the policy-holder to pay- 
to him for the Company. There is more sense than this 
would indicate in the practical business of Life Insurance. 

Numerical Bragging. 

In addition to what has been said of the use made of great 
names in connection with this business^ it is stated by writers 
on Life Insurance, that, " for time out of mind, the practice 
of Numerical Bragging has been by some Companies carried 
to a high pitch of extravagance ; and such Companies rely 
for public favor rather upon the authority of great names 
than upon a full, frank, and conclusive exhibition of their 
affairs." 

It appears that, although safe investments cannot be made- 
in this country at rates of interest higher than from seven to 
eight per cent, per annum, some Life Insurance Companies 
promise, and even guarantee, to pay to policy-holders thirty, 
forty, and as high as fifty per cent, dividends upon the pre- 
miums. Bear in mind that the Company has to pay all the 
current expenses of the business; to pay the losses by death 
during the year, and set apart and retain for investment the 
requisite Deposit for the policy, before it can make any div- 
idends. As the Company can only safely make seven or 
eight per cent, interest upon the funds intrusted to it, these 
enormous dividends to policy-holders, after meeting the above 
obligations, look like numerical bragging. 

Let us see a little further into this enormous dividend. 
In the first place, the expenses of Life Insurance Companies 
are large. Agents' commissions, salaries of officers, travel- 
ing expenses, taxes, printing, rents, stationery — these, and 
other expenses, have to be paid in cash. The losses that 
occur during the year, by death, must be paid in cash. The 
expenses and the losses b}^ death are paid by the Company; 

BUT this is done WITH THE MONEY OF THE POLICY-HOLDERS. 

The Deposit is a specific amount, determined by accurate 
arithmetical calculation ; this amount must be in the hands 
of the Company, and held securely invested at a certain rate 
of interest, and this interest regularly compounded every 
year, in order to enable the Company to pay its policies at 



Life Inswrance. 99 

maturity in money. If the Company has in its hands the 
requisite Deposit, it is solvent. If it has not on hand, and 
securely invested at the fixed table rate of interest, the full 
amount requisite for the Deposit, the Company cannot pay 
its policies in money at maturity. Of course the Company 
must retain the requisite Deposit for each and every one of 
its outstanding policies ; must pay current expenses ; must 
pay the losses that occur by death, each year, of a certain 
number of policy-holders ; and as the Company can only 
make seven or eight per cent, by safe investments of the 
funds intrusted to it by the policy-holders, the Enormous 
Dividends so much talked of look like Numerical Bragging. 

The Deposit is not Cash Capital. 

When a purely Mutual Company advertises $12,000,000 
CASH CAPITAL, and an examination of the official reports 
show that nearly $10,000,000 of this Cash Capital is the 
Deposit — an accrued liability — a debt; and that the Compa- 
ny would be insolvent if it had not the means on hand to 
pay this debt, nearly $1,000,000 of the Cash Capital proves 
to be premium notes paid to the Company by policy-holders 
in lieu of money; and something more than $1,000,000 
turns out to be Surplus, belonging to, but withheld from, 
individual policy-holders. It would seem that the Numeri- 
cal Bragging alluded to by former writers, is still practiced 
by some Life Insurance Companies, when they convert 
$10,000,000 of debt, $1,000,000 in premium notes, and 
$1,000,000 of retained Surplus, into $12,000,000 Cash Capi- 
tal. 

Assets Three Times the Liabilities. 

Some Companies boast of assets amounting to three times 
their liabilities. It should be borne in mind, that when the 
insured pays his annual premium, the Company at once 
becomes liable for the expenses for the following year for 
the Cost of Insurance or losses by death during the year, 
and for the Deposit at the end of the year. Assets amount- 
ing to three times the liabilities of a Company, indicate a 
bad case of numerical b?'aggi7ig ; and even the authority of 



100 J^otes on 

great names cannot long give weight and influence to Com- 
panies that present statements of the character referred to 
above, viz : Enormous Dividends — Purely Mutual Company — 
with $12,000,000 Cash Capital, and assets of a Life Insur- 
ance Company amounting to three times its liabilities. 

No Dividends from Net Peemiums at Net Interest. 

What is called the Net Premium in Life Insurance is a 
matter of direct arithmetical calculation, based upon a 
Mortality Table which gives the law of duration of human 
life when applied to large numbers of mankind. The Mor- 
tality Table is based upon statistical facts. In these calcu- 
lations a rate of interest is taken so low that it may safely 
be assumed that this interest will be realized and regularly 
compounded every year during the existence of the contract 
between the policy-holder and the Life Insurance Company. 
This period is generally for the lifetime of the policy-hold- 
er. 

The net annual premium for each policy, as above deter- 
mined, is enough, and only enough, to pay the losses that 
occur each year by the death of policy-holders, and provide 
the requisite Deposit, at the end of each year, which will 
enable the Company to pay all its obligations at maturity. 
No dividends need be expected then from the net premium at 
net or table interest. 

Policy-holders Should Investigate Certain Points. 

Beyond what the law can do for a policy-holder, it is well 
that he attend, either himself or through a competent person 
on whom he can rely, to several matters bearing on the ques- 
tion of Life Insurance. "Great names" and high business 
qualifications in other professions, are not in themselves suf- 
ficient to conduct Life Insurance successfully. The question 
is, do the officers of the Company comprehend the principles 
upon which the business is founded; and do they give their 
own close personal attention to it. If they do not under- 
stand and closely attend to the business, their high char- 
acter, business capacity, and great names, are a delusion in 
Life Insurance. If the officers are, in every respect, the 



Life Insurance. 101 

right men for this most serious, important, and gigantic busi- 
ness, it is well to look further and inquire closely into the 
terms and conditions of the contract between the Company 
and the policy-holder. These are expressed on the face of 
the policy; and in some Companies are liberal and just; in 
others they are vexatiously and harshly restrictive, not to 
say unjust. It is but a few years since it was the universal 
practice of Life Insurance Companies to appropriate to 
themselves the whole accrued value of a policy in case 
the holder thereof failed on a given day to pay his annual 
premium. 

Suppose that the old man, whose account we followed, 
year by year, from age thirty, to include age ninety-nine, 
had failed to pay his last annual premium : under the rule 
followed by all Life Insurance Companies, only a few years 
ago, his Deposit, amounting to $9,445.70 on a policy of 
$10,000, would have been declared forfeited, and the Com- 
pany would have pocketed this money. Bear in mind that 
this policy had contributed its proportion to pay the losses 
by death of other policy-holders every year; had paid its 
proportion of the yearly expenses, and that $1,652 of this 
old man's money, not including interest, had already gone 
direct into the private possession of the share-holders, with- 
out in the slightest degree touching the matter of his insur- 
ance. Only a few years ago this Deposit, produced entirely 
by the policy-holders' money, would have been confiscated 
by the Company, thus robbing the weak and unfortunate to 
increase the dividends of the strong and rich. There was 
no justification or excuse for this rule of forfeitnie for non- 
payment of premiums « except in the fact that this was a 
condition expressed in the contract. That it cojitinued for 
so long a time to be the universal custom can only be ac- 
counted for by the fact that the principles upon which Life 
Insurance is founded were not thoroughly understood by 
business men. The terms of the contract require atteniion, 
because the quality of the article purchased by the policy- 
holder, i. e. the policy, is directly affected by these terms, as 
well as by the considerations previously mentioned. Now 
we come to the question of price charged. There can be 
no safety or certainty of the payment of policies at matu- 



102 J{otes on 

rity, and, therefore, no real insurance, in case a Company 
charges less than the net annual premium, and enough more 
to cover the expenses. Because, in spite of all we hear 
about large ^^ dividends''^ to policy-holders, arising from the 
"investments^ of premiums, the net annual premium and net 
interest upon it must go to effect the insurance, and the 
expenses must be paid in addition. It appears from this 
view of the case, that a Life Insurance Company may charge 
too little. It may do this and be irretrievably insolvent, and 
still give no external sign of its condition for the lifetime 
of a generation, because, in the first thirty or forty years of 
the existence of a Company, the annual premiums are large- 
ly in excess of the death claims. 

Certainty of the payment of his policy at maturity is 
WHAT every policy-holder WANTS. To insurc this, it is nec- 
essary that the Company should charge enough to enable it 
to meet all its liabilities during the worst year that may 
reasonably be expected to occur during the continuance of 
the contract; and this is generally for a lifetime. There- 
fore, when the mortality is greatest, and the interest on in- 
vestments lowest, and expenses heaviest, the Company must 
have the means of meeting its liabilities. It follows that, 
in favorable years, there will be an over-payment. In case 
this over-payment is all returned to the policy-holder at the 
end of the business of the year, it would seem that it is not 
a matter of vital importance whether the premium is a little 
more or a little less, provided it is enough to make the pay- 
ment of the policy at maturity certain ; and provided further, 
that the business of the Company is managed with ability, 
integrity, and economy ; that the trust is well administered 
in every respect, and the policy is liberal and just in its 
conditions. But if a limited number of share-holders, with 
capital stock, say of $100,000 or $200,000, are by their char- 
ter authorized to appropriate to themselves twenty, or even 
five, per cent, of the surplus arising from the over-payments 
made by a large number of policy-holders, these capitalists 
will make enormous profits out of the funds a man sets 
apart, whilst living, for the protection of his widow and 
orphans from poverty and want that might befall them, in 
case of his early death. On a really Jiourishing business^ 



Life Insurance. 1 03 

two per cent, of the over-payments made by a large num- 
ber of policy-holders, will make " enormous profits " for the 

SHARE-HOLDERS. 

In 1868 an Insurance Company returned to its policy- 
holders $3,000,000 of surplus arising from over-payments 
made during the year by the policy-holders. This was a 
purely Mutual Company. Suppose that a Company, with 
$100,000 of capital stock, succeeds in attaining a like flour- 
ishing business, and that the share-holders of this Company 
have secured to themselves, by special charter, the perpet- 
ual right to appropriate to themselves twenty per cent, of 
all the surplus arising from over-payments made by the pol- 
icy-holders, the " Happy Capitalists " of such a Company 
would receive $600,000 per year in addition to legal inter- 
est upon their $100,000 of full-paid stock. It is hardly nec- 
essary to tell the policy-holder that it is his money that 
produces these "enormous dividends" for the share-holders. 
The "Sacred Fund" set apart by our "friend and neigh- 
bor" during his life, for the purpose of protecting his family 
from poverty and want after his death, is hardly the source 
from which capital should seek to make six hundred per 
cent, per annum. Capital, however, is proverbially aggres- 
sive, and will not hesitate to take twenty per cent, of the 
surplus of the policy-holders, provided the latter agree to it. 
The per cent, of surplus secured to share-holders by their 
charter is a matter upon which no prudent policy-holder 
should fail to inform himself On the other hand, it is not 
always safe to assume that the purely mutual and entirely 
philanthropic basis is necessarily the best reliance in a busi- 
ness sense. Even 'the purely Mutual Companies do not, as 
a rule, distribute all their surplus ; nor do the policy-holders 
in such Companies get this business attended to for them 
for nothing. 

The share-holders of some of the mixed Companies receive 
no other compensation for the use and risk of their capital, 
and their own personal attention to the business, than the 
legal interest earned by that capital. This is rather too 
generous on their part; but it may be that the share-holders 
find sufficient compensation in the honor and consequence 
arising from their being the custodians, and having the per- 



10 Ji^ J^otes on 

manent control and handling, of millions of dollars in trust 
for other people. 

It is believed by many that the purely Mutual Companies 
are defective in an essential particular. They offer no 
adequate inducements for the best business men to become 
'trustees, and to devote the same attention and energy to 
the business of the Company that they would give to their 
own personal and private affairs. It is true that many of 
the purely Mutual Companies have competent officers, and 
give them handsome compensation for their attention to 
the business of the Company; but after all, this is a salaried 
service, which, by business men is not, as a general rule, 
considered as safe a reliance as that of personal ownership 
and money of share-holders staked upon the success of an 
enterprise. 

It is stated by some writers that the plan of compensating 
share-holders, by allowing them legal interest upon a limited 
amount of an additional guarantee fund, formed gradually 
by retaining in the hands of the Company a portion of the 
Surplus, " identified the interests of the stock-holders and policy- 
holders, and thus guarded both, while it tended to reduce largely 
the average of the Company'' s losses and expenses.'^'' 

Insurance Partly on Credit. 

Many persons insist that it is cheaper, safer, and better for 
men to insure their lives partly on credit, than it is to insure 
on the all-cash plan. This note or loan system of Life 
Insurance has strong advocates amongst well-informed in- 
surance writers. But in the long run policy-holders will 
tind there is some delusion about the credit so generously 
proffered and urged upon their acceptance. It is true that 
if a man is certain that he will die soon, and he can get 
$100 worth of insurance for $50 in cash and his note for 
$50, he would do well to take out a policy in a note 
Company, die during the year, and let his heirs receive the 
amount of the policy, less his note for $50; but there are 
many and strong reasons why the system of Note or Loan 
Life Insurance is not advantageous to those who continue 
to renew their policies in such Companies for any great 
length of time. 



Life Insurance. 1 05 

Proprietary or Purely Stock Companies. 

Besides the purely Mutual Companies and the mixed 
Companies, there are Companies conducted on the strictly 
proprietary plan. These purely Stock Companies, in which 
all the surplus belongs to the share-holders, as a general 
rule, charge less premiums than the purely Mutual or mixed 
Companies; but they return no surplus to policy-holders — 
their theory is, that they make dividends to their policy- 
holders IN advance, by charging less premiums. The fact 
is, that dividends to holders of Life Insurance policies are 
simply a return of that part of the annual premium which 
was paid to the Company at the beginning of the year, 
and which, at the end of the business of the year, is found 
not to have been required in paying the expenses, paying 
the losses by death, and providing the requisite Deposit at 
the end of the year. The real dividends in Life Insurance 
— that is to say, real income produced by the. investment of 
capital in business — are made to share-holders, and these, as 
seen above, are sometimes enormous. 

A great deal depends upon the capacity, good judgment, 
close personal attention and integrity of the officers in 
control; but it may be assumed as certain that a correct 
knowledge of the principles upon which the business is 
founded — that is to say, the peculiar arithmetical law% ap- 
plicable to these future and contingent values — should be 
clearly understood ; and it is hardly possible for any man 
to conduct this business successfully without understanding 
clearly the principles upon which the calculations of values 
and the liabilities incurred are based. 

Great names alone will not answer the purpose ; nor will 
numerical bragging command ultimate success. 

Accumulation Necessary in the Earlier Years. 

One of the most striking features in the practical business 
of Life Insurance, as at present generally conducted, arises 
from the fact, that, in the early years of a policy, it is neces- 
sary to accumulate money, in order to meet the demands 
arising in the later years, when the death claims will ^o 
largely exceed the premiums. This causes these Couipa- 



106 Jfotes on 

nies to be the custodians of millions of money ; and all 
these millions must be regularly and safely invested. 

May be Made Safe, but it Needs Watching. 

Life Insurance may, from its peculiar nature, be made, 
perhaps, the safest business known — at the same time, in 
the hands of those ignorant of its principles, or incompetent 
to control, even if they understand it, it can go further 
wrong, and show less evidence for years of its utter insol- 
vency, than any other business ever devised. And whilst 
it may be made the safest, it is a business in which, if it is 
not thoroughly comprehended and strictly guarded, design- 
ing fraud may raise a curtain behind which its worst 
schemes can be carried on free from detection, until such 
time as the death claims exceed the annual premiums ; that 
is to say, for thirty or forty years. 

To fully appreciate this fact, it is only necessary to recall 
the illustration previously given, in which it was seen, that, 
at the end of the thirty-fourth year, nearly $28,000,000 was 
on hand in Deposit, after paying all the death claims that 
had previously matured. This sum, and all the future net 
annual premiums, with compound interest on the whole, 
is required in order to enable the Company to meet its 
liabilities. Suppose that this $28,000,000 had been appro- 
priated to other purposes? This might have been done^ 
and the Company have paid all its losses up to that time, 
and, to external appearance, have seemed all right; and this, 
too, with a real defalcation of $28,000,000. 

One Hundred Thousand Dollars Deposited is of but Little 
Avail in Certain Cascs. 

It is no doubt, upon good grounds, that the law of some 
States requires that $100,000 of capital stock should be paid 
up and deposited with the Treasurer of the State before a 
charter is granted to a Life Insurance Company. 

But it is well that policy-holders and all others interested 
should know the fact that, in a Company with a large num- 
ber of policies of long standing, capital stock of $100,000, 
or even $1,000,000, in the hands of a State Treasurer, would 



Life Insurance. 107 

not secure the payment of premiums in that Company, in 
case the Deposit had been misapplied, lost, or stolen. Cap- 
ital stock in a Life Insurance Company that has succeeded 
in securing a moderate number of policy-holders, is of but 
little avail except in securing for the Company the close per- 
sonal attention of business men to their own private interests. 
But to effect this, it is not considered that it is absolutely 
necessary to give these share-holders an easy opportunity 
for making six hundred per cent, per annum upon their 
stock. 

Custodians of Immense Sums of Money. 

Life Insurance Companies, by their intrinsic nature, must 
become, if moderately successful in securing business, the 
custodians and investors of immense sums of money. The 
location of a Company, as well as the experience, capacity, 
integrity and industry of those who manage the business, is 
a subject for close consideration. But it should be borne 
in mind, that, because ten per cent, is legal interest in one 
State, and six per cent, is the legal interest in another, it 
does not necessarily follow that a Company chartered in the 
latter may not invest its funds in the best safe market just 
as well as the former. 

Agents' Commissions. ^ 

The large per cent, of the premiums paid to agents is an 
item of very heavy expense to Life Insurance Companies. 
And another great expense is the publishing of a large 
amount of what is called " Campaign Literature." It is 
perhaps impracticable fctr the Companies to materially les- 
sen these enormous expenses, so long as the present ex- 
traordinary competition is kept up, and the public are un- 
informed IN regard to the true principles upon which the 
business ought to be conducted. 

If policy-holders had clear and distinct ideas of their own 
in regard to Life Insurance, and would seek for the best 
article at the fairest price, in this business, as they already 
do in regard to their other purchases, the best Companies 
would no doubt be but too glad to abate from their premiums 



108 ' J^otes on 

that portion of the " Loading " which now goes to pay these 
large commissions to agents. And it would no longer be 

PROFITABLE FOR THE AGENTS TO REPRESENT THAT, BY BUYING AND 
PAYING FOR A LIFE INSURANCE POLICY, THE POLICY-HOLDER IS, 
IN ADDITION, MAKING AN INVESTMENT THAT WILL PAY HIM FORTY 

PER CENT, "dividends;" and this, too, after the Company loans 
the policy-holder one third or one half the premium, and 
charges him but six per cent, for the accommodation. 

■ A Large Deposit or Reserve Means a Large Debt. 

The large Companies that have outstanding policies, upon 
which the Deposit has been accumulating for years, are 
supposed by the general public to be immensely rich. And 
man}^, believing that this is really true, find in it sufficient 
reason for the pertinacity and importunity with which they 
are urged by certain parties to insure their lives. The Com- 
panies that have large Deposits are not rich; but they are 
the custodians of immense sums of money, belonging, it is 
true, to others, but invested and handled by the Company. 
This is in itself a matter of immense moment. Add to it 
the enormous cash dividends to share-holders that must arise 
in Companies that have acquired by charter the right to 
appropriate twenty per cent, of the surplus arising from 
over-payments made by the policy-holders, and it is not 
difficult to see and understand why men are pressed to pa- 
tronize their " friends and neighbors." If the Company can 
once attain a " flourishing business," the " Happy Capitalist" 
will get six hundred or seven hundred per cent, yearly divi- 
dends on his stock in addition to the power and other inci- 
dental advantages to the Company, arising from the fact 
that it handles and controls millions \ipon millions of money. 

In the current business of each year, large amounts are 
received from premiums, large amounts are paid out for 
losses by the death of policy-holders, and a vast deal goes 
to pay the ordinary expenses; but in old-established cem- 
panies, that have large numbers of policies that have been 
in force for years, the Trust Fund Deposit is the great item. 
And it is upon the security and safety of this fund that the 
ultimate ability of a Life Insurance Company to pay all 
its policies at maturity mainly depends. 



Life Insurance. 109 

Not Exempt from the Usual Results of Mismanagement in 
Ordinary Business. 

Notwithstanding the accuracy of the theory upon which 
the business of Life Insurance is founded, there are many 
contingencies that may prove fatal to Companies in prac- 
tice; and whilst strict compliance with certain fixed prin- 
ciples and definite rules will always enable a Company to 
pay its policies at maturity, there are many things that 
will, if permitted to occur, bankrupt a Life Insurance Com- 
pany just as certain as a disregard of the peculiar laws gov- 
erning this business will lead the Company ultimately to 
inevitable destruction. These Companies are not exempt 
from the effects produced by dishonesty, fraud, and defalca- 
tion. Moreover, continued lavish expenditures, the selec- 
tion of bad risks by insuring impaired or unhealthy lives, or 
making unsafe investments, will lead to bankruptcy certain. 

There can scarcely be any saying more groundless than 
the statement often heard, that " Life Insurance Companies 
cannot break." And, on the other hand, it is absurd to say, 
that, when well-conducted in every particular, it is impos- 
sible for Life Insurance Companies to comply with all their 
obligations, and pay all their policies at maturity. The 
plain fact of the case is, that Life Insurance Companies can 

BREAK, AND WILL BREAK, UNLESS 'MANAGED WITH SKILL AND INTEG- 
RITY. On the other hand, it is undoubtedly true that the 
BUSINESS OF Life Insurance can be made more safe and more 

SECURE THAN ANY OTHER COMMERCIAL BUSINESS KNOWN AMONGST 

MEN. But honest ignorance cannot^ and designing fraud will 
not, effect this result. Capacity, integrity, industry, skill, and 
sound judgment on the part of those in control, are just 
as essential to success in Life Insurance as they are in all 
other kinds of important business. 

The Principle Upon which Money Values in Life Insur- 
ance ARE Calculated Should be Understood. 

The mere fact that a man can compute interest on money 
will not make him a competent banker, neither will a knowl- 
edge of the formulas and rules be in itself sufliciont to fit a 
man for the important business of Life Insurance. But it 



110 J^otes on 

would be far better to intrust banking to men who cannot 
calculate interest on money, than to intrust Life Insurance 
to those who are not acquainted with the method upon 
which calculations of important^ money-values in this busi- 
ness are based. 

There is danger to all in the doctrine often promulgated by 
agents^ that Life Insurance business can be better conducted 
by men who do not understand the " art of calculating these 
values " than by those who do understand the principles 
upon which alone this business can be safely conducted. 
Those who talk in this way are, generally speaking, forty 
PER CENT. DIVIDEND MEN, who proposc to Icud ouc third or 
one half the premium to the policy-holder at six per cent. ; 
and promise him forty per cent, dividend per annum upon 
the whole amount of the premium. The same persons 
generally style the money of the policy-holder that is held 
by the Company in trust for the purpose of enabling it to 
pay the policy at maturity, cash capital ; or, at least, an- 
nounce millions of assets, and are silent about these assets 
being a deposit debt, held by the Company in trust for other 
people. 

Companies not so Rich as Some People Suppose. 

It is often urged by intelligent business men who are not 
acquainted with the real nature of the Trust Fund Deposit 
(or "Reserve"), that the Life Insurance Companies have 
made vast sums of money, and that the policy-holders must 
have furnished the money that made these Companies so 
immensely rich. This view of the subject has been already 
answered by the statement that this Deposit does not belong 
to the Company, but is, in fact, the money of the policy- 
holder, held by the Company in trust. 

Life Insurance Companies Great Money Lenders. 

It is often urged, too, that Life Insurance Companies are 
absorbing a very large portion of the currency of the 
country ; and many persons seem to apprehend that this 
will result in extraordinary scarcity of money. But it must 
be remembered that Life Insurance Companies are com- 
pelled to keep their funds constantly invested ; they are, 



Lif& Insuranfe. Ill 

thei-efore, forced to be lenders of money ; and, as a general 
rule, they are more careful about the character of their 
securities than anxious to realize exorbitant rates of inter- 
est. 

In Practice, Payment not Postponed to the End of the Year. 
Although in theory the amount of a policy is not due until 
the end of the policy year within which the insured may die, 
it is usual for Life Insurance Companies, in practice, to pay 
the policy within from thirty to ninety days after proof of 
the death of the policy-holder. 

Policy-Holder Assumed to be Aged an Exact Number of 

Whole Years. 

It is usual to assume that a person who applies for insur- 
ance is exactly a given number of years old. The Mortality 
Tables and the calculations are based upon whole years ; 
and the age is taken to be the w^hole number of years 
nearest to the real age. For instance, if the real age of 
a person was thirty years and five months, he wou4d be 
considered thirty years old ; but if the real age was thirty 
years and seven months, he would be taken as thirty-one 
years old. 

Extreme Haste in Advertising the Payment of One Policy. 

There is a great deal of what, in common parlance, is 
called " clap-trap," resorted to by some Life Insurance Com- 
panies and agents. Besides what has been previously stated 
in reference to the borrowing of great names, the enormous 
dividends to policy-holders, and numerical bragging in gen- 
eral, attention is here called to the extreme haste with which 
some Companies and agents rush the announcement into 
the newspapers, hurriedly advertising the fact that they have 
PAID A POLICY at maturity ; and seem to offer an isolated fact 
of this nature as proof positive that the Compan}^ will pay 
all claims that may hereafter mature against it. What 
would be thought of a bank that took especial pains to 
herald in the newspapers that it had paid one of its obliga- 
tions, and called upon the community to accept this as 
proof that it was solvent and would stand so forever? 



112 yXotes on 

DlFFERZVCE BeT^VEEN THE GevEEAL LaW TPOy WHICH THE Rl5K 

15 Based ly Life IysrsA>XE. avl the ^Xatuee of the Risk 
ly FniE IvsuEAycE. 

As previously stated, more than once, in these ZSotes, the 
general law regulating the duration of human life has been 
very accurately determined. Upon this law. and an as- 
sumed safe rate of interest, all Life Insurance calculations 
are based. Xotice the contrast between this and the data 
upon which the calculations or estimates in the business 
of Fire Insurance are founded. It was stated by the Xa- 
tional Board of Fire Underwriters, in a report dated 1S6S, 
that, "as a icholey the business (of Fire Insurance; is absolutely 
without that chart of experience, furnished only by combined re- 
sults, carefully^ noted and preserved." (See Xew York Insurance 
Report. 1S6S, page 7.) Xotwithstanding this, nearly every 
prudent basiness man insures his property against destruc- 
tion by fire : and this. too., when there is a strong probabil- 
ity that his property will never be destroyed by this cause. 
It is certain that all men must die The business of Life 
Insurance can be made safe : and yet there are xery many 
men who have not insured their lives, and have no present 
intention to do so. In many cases this probably arises from 
a belief that the system itself does not rest on principles 
and laws that are certain and stable ; and in other cases 
it perhaps arises from some apprehension that the system 
may not be fairly and honestly administered. 

FUETEER ALLUSloy TO TZE GevEEAL LaW OF DuEATlOy OF 

FIrMAy Life — Melical Examivees. 

Ordinary* business, in times of panic or extra stringency 
in the money market, is liable to a sudden strain upon its 
resources that often proves fatal ; whereas, in Life Insur- 
ance, particularly in those Companies that have large num- 
bers of policy-holders judiciously selected and distributed, 
experience has proved that the general law of duration of 
human life holds true with remarkable regularity : and for 
this reason Life Insurance Companies are to a great desrree 
exempt from those sudden and extreme demands upon their 
resources, which are at many periods so fatal to ordinarv 
commercial business. 



Life Insurance. 113 

But the general law governing the duration of human life 
will be of little or no avail in case a Life Insurance Com- 
pany accepts risks upon impaired or diseased lives ; and 
Companies that have only a small number of policy-holders 
will always be, to some extent, liable to a number of losses 
not in accordance with the general law of duration of 
human life ; because this law only applies to large num- 
bers, not to a single individual, or to a small number of 
individuals. Much of the success of Life Insurance Com- 
panies depends upon the skill and integrity of the Medical 
Examiners. 

Lawyers — Professors of High Schools — Accountants. 

Lawyers should understand the true principles of Life 
Insurance, because $2,000,000,000 was never yet staked 
upon any one business without the subject being sooner or 
later brought into the courts. 

Professors of High Schools should understand the principles 
of Life Insurance, because this business has already attained 
magnitude, such that every intelligent, educated man in the 
country ought to have a correct knowledge of its nature 
and bearing upon the general welfare. 

Educated accountants, book-keepers, clerks, and computers, 
ought to understand the manner in which these accounts 
are made, and the values calculated. They may all rest 
assured that there is nothing in the theory, the principles, 
or in the method of calculating values in Life Insurance, 
that is not within the easy comprehension of men who have 
a thorough knowledge of the single rule of three, and a 
mere acquaintance with the simplest principles of elementary 
algebra. The intelligent, general business sense of the 
country should be informed definitely what Life Insurance 
is, and what it is not; and it is hoped that the foregoing 
Notes may tend, in some degree, to promote this end. 
8 



"In a body of lives of the same age, all selected as healthy from the general 
mass of mankind, it is obvious that the rate of mortality must be considerably 
less for the first ten or twenty years after selection^ than amongst those from whom 
they are thus chosen; as, however, these selected lives advance in age, their 
general health, and the rate of mortality amongst them, ■will naturally ap- 
proximate to the common standard." — Morgan. 

"One is struck with the fact that assured lives are, for some time after selec- 
tion, much better than the community at large, hut that after awhile they become 
much worse. This can arise from no other cause than the selection which the 
assured exercise against the Company by dropping policies on healthy lives, and 
retaining those on lives which have become bad or doubtful." — Higham. 

*' Those persons will be most for flying to these establishments who have feeble 
constitutions, or are subject to distempers which they know render their lives 
particularly precarious; and it is feared that no caution will be sufficient to pre- 
vent all danger from hence.'^ — Dr. Price. 

" The necessity of the valuation to an effective supervision, arises from the 
peculiar nature of the business of Life Insurance. In this peculiarity lies its 
greatest danger. The opportunity for fraud or fatal error. Life Insurance 
reverses the laws which govern all other commercial enterprises and invest- 
ments. In the latter the expenditure comes first, and the profits, if any, come 
afterwards. In the first years of a Life Insurance Company, its treasury over- 
flows with the incoming premiums, whilst its liabilities are postponed for the 
lifetime of a generation. For more than thirty years it furnishes a constant 
margin for plunder or perversion of its funds, while its ultimate failure, though 
certain if the opportunity is improved, is still remote. Unless its condition is 
probed by some decisive test, it exhibits no necessary symptoms of its insol- 
vency until the claims by death begin to equal or exceed the premium receipts ; 
and this period will not ordinarily be reached until nearly forty years from its 
start. If it can once be fairly believed that there is no mystery surrounding 
the process technically called valuation, too deep for ordinary ken, its reasons 
and importance may be better, or at least more generally, understood." 

William Babnes. 



Life Insurance, 115 

GENERAL SUMMARY OF FORMULA AMD ARITHMETICAL RULES 
FOR LIFE INSURANCE NET CALCULATIONS. 

Net SmtJLE Premium to Insure One Dollar for Whole Life. 

Rule. — Subtract v from 1 ; then multiple/ by A^, as found in 
the tables^ lor, which is the same thing, multiply by ~\; and sub- 
tract the product from one doll^ar; the remainder is the net single 
premium that will insure one dollar for whole life at the age x. 

Note.— iJ, in all cases, is equal to 100, divided by 100, plus 
the table or net rate of interest, 

Net Annual Premium to Insure One Dollar for Whole 

Life* 

l^vhi^.— Subtract v from 1; subtract this remainder from the 
quotient obtained by dimding 1 by A^.. The result is the net 
annual premium that will^ at age x, insure one dollar for whole 
life. 

To Determine the Amount of the Trust Fund Deposit (or 
Reserve) at the End of n Years on a Whole Life Policy 
FOR One Dollar, Taken Out at the Age x. 

(T. F. D.).+.=l-%^. 

Rule. — First find from the table tne value of large A at the 
age x-\-n ; then the value of large A at the age x; divide the 
former by the latter, and subtract the result from one dollar. 

General Formula for Calculating the Trust Fund Deposit 
(or Reserve) at the End of n Years on any Kind of 
Policy for One Dollar, Taken out at Age x. 

(T. F. D.).+.=».+,_i((T. F. D.).+.-,+aP.-c,+„_, 



Rule. — Calculate aV^ by the formula applicable to the case; 
add to tJiis the Deposit (or Reserve) for the year n — 1; then sub- 
tract from this sum c^j^^_^ {obtained from the table); and multiply 
the remainder by m^^.,i_i {obtained from the table) ; the rcsxdt is 
the Deposit {or Reserve) for a policy of one dollar at the end of 
n years from its date. 



116 J{otes on 



TERM INSURANCE. 

Net Single Premium that will Insure One Dollar for n 

Years. 



Rule. — From the tables find the value of "N at the age x ; then 
find the value of ISi at the age x-]-n; subtract the latter from the 
former, and multiply the remainder by v. Then find from the 
tables the value of 'N at the age x-}-l, and subtract from this 
the value of 'N at the age x^n-\-l. Now subtract this remainder 
from the amount obtained by multiplyiug v by (N^ — N^+n); and 
divide this remainder by D at the age x (obtained from the table) ; 
the result is the net single premium sought. 

Net Annual Premium that will Insure One Dollar for n 

Years. 

Rule. — Find from the table the value of N at the age x-}-!; 
subtract from this the value of N at the age x-\-n^l. Then 
divide this remainder by the remainder obtained by subtracting N 
at the age xA^n from N at the age x. Subtract the result of this 
division from v, and we have the net annual premium that will, at 
age X, insure one dollar for n years. 

Net Single Premium for* an Endowment of One Dollar at 
the End of n Years. 

]>y[oTE. — This is endowment pure and simple, and does not in- 
clude insurance. 



EJ. 



D 



x-\-n 



Rule. — Find from the table the value of D at the age x-\-n ; 
and divide this by the value of D at the age x. 

Net Annual Premium for an Endowment of One Dollar at 
the End of n Years. 



■"•icln ■'-^x . ^^x-\-n 



Life Insurance, 11 7 

Rule. — Find from the table the value of D at the age x-^-n ; 
and divide this hy the difference between N at the age x and N 
at the age x-\-n. 

Net Single Premium for Insurance of One Dollar for n 
Years, and an Endowment of One Dollar at the End 
of n Years. 

I ^x 

Rule. — Find from the table the value of N at the age x ; sub- 
tract from this the value of N at the age x-\-n ; multiply the re- 
mainder by V, and add this product to the value of N at the age 
x-\-7i; then subtract from this sum the value o/N at the age x-]-! ; 
divide the remainder by D at the age x, and we have the net 

single premium that will, at age X, INSURE ONE DOLLAR FOR 

n years, and at the same time provide for an endowment of 
one dollar to the insured at the end of u years, in case 
he is alive at that time. a 

Net Annual Premium for Insurance and Endowment of One 
Dollar as Above. 

(.P+E)J. _^^_ N,.^-N.^. 
A.|, N -N,^,. • 

Rule. — Find from the table the value of "N at the age x -{-I; 
subtract from this the value of N at the age x-\-n. Then di- 
vide this remainder by the difference between N at the age x, and 
N at the age x^n. Subtract this result from v, and we have 
the net annual premium IN this case. 

The Net Annual Premium for n Years that will Insure 
One Dollar for Whole Life is Expressed by — 

Rule. — Find from the tables the value of N at the age x; vml- 
tiply this by (1 — v) ; subtract the product froni D at the age x; 
and divide the remainder by the difference between N at the age 
Xy and N at the age x-\-n. This rule gives the net annual 
premium that will, if paid annually for n years (provided 



118 J^oies on 

the insured is alive to make the payment), insure one dollar 
to be paid to the heirs of the insured at the end of any year 
in which he may die. 

Explanation of the Manner in which the Tabkes Inserted 
IN THIS Work were Constructed. 

Table I. The Mortality Table adopted in the foregoing 
** Notes," is the result of experience in seventeen leading 
English Companies; and is generally called "The Actua- 
ries'." 

The net interest is taken at four per cent, per annum, v, 
in this case, is equal to 100, divided by 104; or $0.961538. 

The quantity we have represented by A,; is equal to a 
fraction, the numerator of which is represented by N^, and 
the denominator by D^,. 

By reference to page 27, it will be seen that at age ninety- 
nine the numerator and denominator are each represented 
l>y ^^%9- The value of A^ is, therefore, in this particular 
case, equal to unfty. It is, however, essential that the par- 
ticular value of v^%^ shall be determined /gg is equal to one, 
because this is the number of persons living at that age 
according to the Table of Mortality we are now using. 
The question, then, is simply, what will $0.961538 become 
w^hen raised to the ninety-ninth power? This multiplica- 
tion is a matter of simple arithmetic. By referring to page 
twenty-eight, it will be seen that, at age ninety-eight, 

Referring to the previous multiplication, we find v to the 
98th power. 

Multiply this by the number living at age ninety-eight, 
which, by the Table of Mortality we are now using, is four, 
and we have the numerical value of v^%^. This is the first 
term of the numerator, and it is at the same time the denom- 
inator of the fraction which gives the value of A at the age 
ninety-eight. Add to this first term of the numerator the 
value of t5^%9, just before obtained, and we have the numer- 
ator of A at the age ninety-eight. And so working back 
an age one year less each successive time, we find the first 
term of the numerator; and this, in every case, gives the 



Life Insurance, 119 

denominator of that age. The second term of the numera- 
tor will be the first term of the numerator, and also the 
denominator of an age one year greater ; and thus, after 
finding the first term of the numerator at any age, it not 
only gives the denominator for that age, but we have only 
to add this to the numerator of the age one year greater 
in order to obtain the numerator at the age sought for. 
Continuing in this way, diminishing one year at a time, 

N 
we will finally obtain Aio=~^j ^^ which the first term of 

the numerator and the whole denominator will be expressed 
by v^\q' The second term of this numerator is the first term 
of the numerator of an age one year greater ; the third 
term of the numerator is the first term of the numerator of 
an age two years greater, and in like manner to age nine- 
ty-nine. 

Op a _ Dio+Dii+Di2+Di34- to Dg, 

-L'lO 

Before giving the following table of numerators and 
denominators and the resulting values of A^ at the different 
ages, all of which have been calculated by the above method, 
only using a table of logarithms to find the different powers 
of V, instead of actually performing the multiplications in- 
dicated ; the following extract is given as a matter of 
interest connected with these tabulated values of the nu- 
merators and denominators of the fractions that express the 
value of A at the diff'erent ages. Professor Elizur Wright 
says, in a foot note, pages 362, 363 of his official reports, 
published in 1865, alluding to the device of multiplying 
both the numerator and denominator of A^ by v to the x 
power: "This happy thought occurred quite independently 
to two mathematicians, one an erudite Danish or German 
Professor, and the other an unlearned Englishman, a poor 
farmer's boy, in fact. John Nicholas Tetens, Professor of 
Mathematics in the University of Kiel, published the method 
in 1785. None of the mathematicians or Life Insurance 
Actuaries of England were the wiser for this, but plodded 
on, making, in a far more laborious way, the computations 
demanded in practice, till George Barrett, a wholly self- 
taught and rather poverty-stricken computer, astonished 



120 Jfotes on 

them about 1811 by a vast mass of tables computed on a 
method of his own, which was the one above explained. 
He had commenced his labors on these tables about the 
time when Tetens published in German, but it is quite cer- 
tain that he was as ignorant of the existence of that publi- 
cation, not knowing either German or French, as were all 
the English Professors. It took the savans of the Royal 
Society several years, after their attention was called to it, 
to recognize the value of the discovery. But after the 
death of poor Barrett, in 1821, it soon ranked among the 
grand English discoveries in the field of useful knowledge, 
under the name of a slight improver, Griffith Davies. A 
quarter of a century rolled away, and then it was discov- 
ered by the scientific Englishman that this wonderful prac- 
tical discovery had been for sixty-five years in their own 
libraries, in German, without their knowing it 1 In the 
standing feud between genius and culture, this is a feather 
in the cap of the former." 

Without further allusion to " the famous columnar method 
OF Tetens and Barrett," we will now refer to the column 
in TABLE I, headed c^. This represents the actual risk 
on one dollar for one year at the age x, and is express- 
ed by V — ^. 4 being the number living at the age x, and 

d^ being the number of deaths between the age x and the 
age x-\-\. For example: at age forty, c^ = 0.961538, mul- 
tiplied by the number of deaths between the age forty and 
age forty-one, which, by the Actuaries' Table of Mortality, 
is 815; and the product divided by the number living at the 
age forty, according to the same table, this number is 78,653. 
The result is c^ at age forty, is equal to 0.009963. 

The values placed in the column headed u^ are obtained 
by dividing the ratio of interest by unity, minus the ratio of 
interest, multiplied by c^. If we call the ratio of interest r, 
then ri;=$l, and r is equal to unity, divided by v. When 
the rate of interest is four per cent., v is equal to $0.961538, 
and r is equal to 1.04. The formula which expresses the 

r 

value of u^ is, u^= —- . 

1 — rc^ 



Life Insurance. 121 

TABLE II is entirely similar to Table I, except in the rate 
of mortality and the rate of net interest. In Table II the 
American experience rate of mortality, and four and one 
half per cent, net interest, is taken as the basis of the cal- 
culations, instead of the Actuaries' rate of mortality and four 
per cent, interest, which was assumed as the basis of the 
calculations in Table I. The latter is in conformity with 
the law of some of the States, and the former of some other 
States. Many of the States have no laws regulating the 
rate of mortality or the rate of net interest upon which 
these calculations are to be made. 

TABLE III shows the actual value of the risk on $1,000 
for one year at any age, the net single premium to insure 
$1,000 for whole life at any age, the net annual premium to 
insure $1,000 for whole life at any age, and the manner of 
determining the Trust Fund Deposit at the end of any pol- 
icy year. 

In Table IV the rate of mortality is the American expe- 
rience, and four and one half per cent, interest; otherwise, 
it is similar to Table III. 

None of these are what are technically called VALUA- 
TION TABLES. The latter are very voluminous ; and be- 
sides taking a great deal of time to construct, are very ex- 
pensive, and only useful in large offices where an immense 
number of policies have to be promptly valued. 

There is reason to believe that a large proportion of the 
600,000 policy-holders in this country, whose lives are now 
insured for more than $2,000,000,000, are nearly as ignorant 
of the peculiar principles upon which the calculations of 
important money-values in this business are based, as the 
writer was but a few months since, when the subject was 
first brought to his attention. I am satisfied that their inter- 
ests demand that more of their number should comprehend 
the nature and peculiarities of the business in which their 
money is invested, and I have, therefore, endeavored to 
explain the subject to them. The foregoing Notes were 
intended, however, to be suggestive ; it having been no part 
of my purpose or desire to force conclusions up^on the mind 
of any one. On the contrary, I have endeavored to give 
to the reader who was not previously acquainted with the 



1^^ Kotes on 

theory and principles of this business the means by which 
to judge for himself; and form his own conclusions in refer- 
ence to the principles, the data, the formula, and rules used 
in calculating money values in Life Insurance. I have 
taken the trouble to write and publish these " Notes," with 
the hope that they may afford to the uninitiated the assist- 
ance I would have been glad to receive when I commenced 
to investigate this subject. I agree w^th the author, w^hose 
opinion is quoted on the first page of this book, that Life 
Insurance^ like all good things, prospers in light rather than in 
darkness. If further information or better " light" is required 
than that which the foregoing" Notes" furnish, I respectfully 
refer all inquirers to the learned writings of authors expe- 
rienced in this business. 

The calculations for the arithmetical examples, introduced 
for the purpose of illustration in the preceding Notes, were 
made from printed tables, in the published Reports of the 
Insurance Commissioner of Massachusetts, Professor Elizur 
Wright. The manuscript of these Notes was placed in the 
hands of three gentlemen who were entirely unacquainted 
with the subject of Life Insurance. After careful reading of 
the manuscript, they kindly offered to assist me in furnish- 
ing an additional arithmetical example for illustration, viz: 
the 'tables hereto appended. The labor proved to be less 
than was anticipated. The four tables annexed were made 
under my general supervision and direction by Major Henry 
T. Stanton, and Professors John A. Monroe and William 
S. Smith, all at present residents of Frankfort, Kentucky. 



WJi, 



.Kotes on 
Table I. 
Actuaries' Rate of Mortality^. \ v = 



4 PER CENT. 



Age. 


3 


5^ 

o . 


10 


100,000 


676 


11 


99,324 


674 


12 


98,650 


672 


13 


97,978 


671 


14 


97,307 


671 


15 


96,636 


671 


16 


95,965 


672 


n 


95,293 


673 


18 


94,620 


675 


19 


93,945 


677 


20 


93,268 


680 


21 


92,588 


683 


22 


91,905 


686 


23 


91,219 


690 


24 


90,529 


694 


25 


89,835 


698 


26 


89,137 


703 


27 


88,434 


708 


28 


87,726 


714 


29 


87,012 


720 


30 


86,292 


727 


31 


85,565 


734 


32 


84,831 


742 


33 


84,089 


750 


34 


83,339 


758 


35 


82,581 


767 


36 


81,814 


776 


37 


81,038 


785 


38 


80,253 


795 


39 


79,458 


805 


40 


78,653 


815 


41 


77,838 


826 


42 


77,012 


839 


43 


76,173 


857 


44 


75,316 


881 


45 


74,435 


909 


46 


73,526 


944 


47 


72,582 


981 


48 


71,601 


1,021 


49 


70,580 


1,063 


50 


69,517 


1,108 


51 


68,409 


1,156 


52 


67,253 


1,207 


53 


66,046 


1,261 


54 


64,785 


1,316 


55 


63,469 


1,375 



N. 



1381771. 

1314215. 

1249696. 

1188079. 

1129236. 

1073044. 

1019385. 
968148.8 
919228.0 
872520.9 
827930.6 
785364.3 
744733.5 
705953.7 
668943.8 
633626.5 
599927.9 
567777.1 
537106.7 



507852. 

479951. 

453346. 

427979. 

403797. 

380749. 

358785. 

337858. 

317922. 

298935.6 

280855.8 

263643.5 

247261 .0 

231671.7 

216841.2 

202736.3 

189326.6 

176583.5 



164480, 

152991, 

142094, 

131765, 

121983, 

112727.9 

103978.49 
95716.59 
87924.14 



D. 



67556.41 
64519.00 
61616.50 
58843.05 
56192.36 
53658.53 
51236.49 
48920.87 
46707.08 
44590.30 
42566.30 
40630.72 
38779.80 
37009.94 
35317.30 
33698.62 
32150.75 
30670.37 
29254.63 
27900.52 
26605.42 
25366.62 
24181.75 
23048.31 
21964.17 
20927.30 
19935.51 
18986.94 
18079.83 
17212.24 
16382.56 
15589.23 
14830.58 
14104.81 
13409.73 
12743.15 
12103.40 
11488.46 
10897.30 
10328.75 
9781.92 
9255.77 
8749.40 
8261.90 
7792.45 
7340.54 



20.453589 
20.369426 
20.281841 
20.190643 
20.095900 
19.997641 
19.895670 
19.790097 
19.680699 
19.567511 
19.450379 
19.329324 
19.204165 
19.074705 
18.940964 
18.802746 
18.659844 
18.512235 
18.359717 
18.202238 
18.039651 
17.871755 
17.698454 
17.519627 
17.335037 
17.144314 
16.947552 
16.744273 
16.534203 
16.317213 
16.092936 
15.861014 
15.621216 
15.373564 
15.118590 
14.857127 
14.589578 
14.316984 
14.039404 
13.757163 
13.470321 
13.179206 
12.884072 
12.585301 
12.283247 
11.977884 



1.04708 
1.04710 
1.04713 
1.04717 
1.04722 
1.04727 
1.04733 
1.04740 
1.04747 
1.04755 
1.04764 
1.04773 
1.04782 
1.04793 
1.04803 
1.04814 
1.04827 
1.04839 
1.04854 
1.04868 
1.04884 
1.04900 
1.04918 
1.04936 
1.04955 
1.04975 
1.04996 
1.05017 
1.05040 
1.05064 
1.05089 
1.05116 
1.05146 
1.05183 
1.05231 
1.05286 
1.05353 
■1.05425 
1.05504 
1.05590 
1.05684 
1.05788 
1,05901 
1.06024 
1.06156 
1.06303 



,006500 
,006525 
,006550 
,006585 
,006630 
,006677 
,006733 
,006791 
,006860 
,006929 
,007010 
,007093 
,007177 
,007273 
,007371 
,007471 
,007583 
,007698 
,007826 
,007956 
,008101 
,008248 
008410 
,008576 
,008746 
,008931 
,009122 
,009314 
,009525 
,009741 
,009963 
,010204 
,010476 
,010818 
,011247 
,011742 
.012345 
,012996 
,013711 
,014482 
,015326 
,016248 
,017257 
.018359 
.019532 
.020831 



Life Insurance. 
Table I — Continued. 
Actuaries' Rate of Mortality. \ v = 

4 PER CENT. 



1^5 



100 
104* 





> 


. 












Age. 


a'" 
52; 




N. 


D. 


A. 


^x 


Cx 


56 


62,094 


1,432 


80583.60 


6905.30 


11.669819 


1.06462 


.022237 


57 


60,658 


1,497 


73678.30 


6486.16 


11.359310 


1.06632 


.023730 


58 


59,161 


1,561 


67192.14 


6082.77 


11.046306 


1.06819 


.025371 


59 


57,600 


1,627 


61109.37 


5694.50 


10.731297 


1.07023 


.027160 


60 


55,973 


1,698 


55414.87 


5320.81 


10.414743 


1.07254 


.029169 


61 


54,275 


1,770 


50094.06 


4960.96 


10.097652 


1.07506 


.031357 


62 


62,505 


1,844 


45133.10 


4614.59 


9.780500 


1.07786 


.033770 


63 


50,661 


1,917 


40518.51 


4281.27 


9.464133 


1.08090 


.036384 


64 


48,744 


1,990 


36237.24 


3960.84 


9.148877 


1.08427 


.039255 


65 


46,754 


2,061 


32276.40 


3653.01 


8.835565 


1.08796 


.042386 


66 


44,693 


2,128 


28623.39 


3357.68 


8.524752 


1.09199 


.045782 


67 


42,565 


2,191 


25265.71 


3074.81 


8.216999 


1.09644 


.049494 


68 


40,374 


2,246 


22190.90 


2804.37 


7.912971 


1.10126 


.053490 


69 


38,128 


2,291 


19386.53 


2546.50 


7.613010 


1.10648 


.057776 


70 


35,837 


2,327 


16840.03 


2301.43 


7.317203 


1.11222 


.062436 


71 


33,510 


2,351 


14538.60 


2069.22 


7.026126 


1.11847 


.067460 


72 


31,159 


2, .-562 


12469.38 


1850.05 


6.740022 


1.12530 


.072889 


73 


28,797 


2,358 


10619.33 


1644.04 


6.459289 


1.13275 


.078734 


74 


26,439 


2,339 


8975.29 


1451.37 


6.184012 


1.14093 


.085065 


75 


24,100 


2,303 


7523.92 


1272.086 


5.914628 


1.14988 


.091885 


76 


21,797 


2,249 


6251.833 


1106.274 


5.651250 


1.15965 


.099211 


77 


19,548 


2,179 


5145.5595 


953.9711 


5.392799 


1.17047 


.107182 


78 


17,369 


2,092 


4191.5884 


815.0313 


5.142853 


1.18242 


.115812 


79 


15,277 


1,987 


3376.5571 


689.2936 


4.898572 


1.19549 


.125062 


80 


13,290 


1,866 


2687.2635 


576.5777 


4.660720 


1.20987 


.135006 


8i 


11,424 


1,730 


2110.6858 


476.5601 


4.428990 


1.22560 


.145611 


82 


9,694 


1,582 


1634.1257 


388.8384 


4.202583 


1.24282 


.156917 


83 


8,112 


1,427 


1245.2873 


312.8677 


3.980235 


1.26200 


.169146 


84 


6,685 


1,268 


932.4196 


247.9139 


3.761058 


1.28344 


.182383 


85 


5,417 


1,111 


684.5057 


193.1634 


3.543653 


1.30833 


.197207 


86 


4,306 


958 


491.3423 


147.6409 


3.327938 


1.33759 


.213923 


87 


3,348 


811 


343.7014 


110.3786 


3.113829 


1.37246 


.232917 


88 


2,537 


673 


233.32281 


80.42416 


2.901187 


1.41549 


.255071 


89 


1,864 


545 


152.89865 


56.81705 


2.691079 


1.46972 


.281137 


90 


1,319 


427 


96.08160 


38.65844 


2.485382 


1.53785 


.311279 


91 


892 


322 


57.42316 


25.13801 


2.284493 


1.62751 


.347103 


92 


570 


231 


32.28515 


15.44570 


2.090236 


1.74867 


.389676 


93 


339 


155 


16.83945 


8.83281 


1.906467 


1.91609 


.439641 


94 


184 


95 


8.006643 


4.609818 


1.736867 


2.15011 


.496446 


95 


89 


52 


3.396825 


2.143990 


1.584347 


2.50162 


.501798 


96 


37 


24 


1.252835 


.857040 


1.461816 


2.95999 


.623701 


97 


13 


9 


.395795 


.289540 


1.366978 


3.37999 


.665680 


98 


4 


3 


.106255 


.085063 


1.240384 


4.15999 


.721154 


99 


1 


1 


.020592 


.020592 


1.000000 




.9G1538 



1^6 



J^otes on 
Table II. . 
American Experience Rate of Mortality. \ v 



100 



104.5 



4>i PEK CENT. 



1 ^ 

S "^ 
p 


^ as 


100,000 


749 


99,251 


746 


98,505 


743 


97,762 


740 


97,022 


737 


96,285 


735 


95,550 


732 


94,818 


729 


94,089 


727 


93,362 


725 


92,637 


723 


91,914 


722 


91,192 


721 


90,471 


720 


89,751 


719 


89,032 


718 


8-8,344 


718 


87,596 


718 


86,878 


718 


86,160 


719 


85,441 


720 


84,721 


721 


84,000 


723 


83,277 


726 


82,551 


729 


81,822 


732 


81,090 


737 


80,353 


742 


79,611 


749 


78,862 


756 


78,106 


765 


77,341 


774 


76,567 


785 


75,782 


797 


74,985 


812 


74,173 


828 


73,345 


848 


72,497 


870 


71,627 


896 


70,731 


927 


69,804 


962 


68,842 


1,001 


67,841 


1,044 



N, 



1214144.06 
1149751.29 
1088592.96 
1030508.12 
975343.79 
922954.55 
873202.12 
825955.57 
781089.94 
738486.41 
698032.49 
659621.22 
623150.91 
588525.23 
555652.58 
524445.85 
494822.18 
466702.78 
440013.03 
414681.95 
390642.02 
367829.27 
346182.84 
325644.85 
306160.43 
287077.60 
270146.88 
253521.14 
237755.95 
222808.94 
208640.17 
195211.51 
182486.98 
170432.25 
159014.89 
148204.10 
137970.87 
128287.62 
119128.49 
110468.94 
102285.96 
94557.98 
87264.70 



D, 



64392.77 
61158.33 
58084.84 
55164.33 
52389.24 
49752.43 
47246.55 
44865.63 
42603.53 
40453.92 
38411.27 
36470.31 
34625.68 
32872.65 
31206.73 
29623.67 
28119.40 
26689.75 
25331.08 
24039.93 
22812.75 
21646.43 
20537.99 
19484.42 
18482.83 
17530.72 
16625.74 
15765.19 
14947,01 
14168.77 
13428.66 
12724.53 
12054.73 
11417.36 
10810.78 
10233.23 
9683.250 
9159.133 
8659.541 
8182.981 
7727.976 
7293.276 
6877.731 



18.855286 
18.799583 
18.741431 
18.680699 
18.617252 
18.550916 
18.481821 
18.409536 
18.333924 
18.255002 
18.172596 
18.086522 
17.996738 
17.903180 
17.805534 
17.703614 
17.597187 
17.486261 
17.370486 
17.249715 
17.123883 
16.992622 
16.855734 
16.713063 
16.564535 
16.409932 
16.248753 
16.081066 
15.906540 
15.725304 
15.536887 
15.341389 
15.138245 
14.927418 
14.708902 
14.482603 
14.248399 
14.006573 
13.756972 
13.499817 
13.235924 
12.965084 
12.638010 



1.052885 
1.052914 
1.052942 
1.052971 
1.052999 
1.053038 
1.053066 
1.053096 
1.053136 
1.053177 
1.053220 
1.053274 
1.053328 
1.053382 
1.053439 
1.053496 
1.053564 
1.053636 
1.053709 
1.053793 
1.053881 
1.053975 
1.054072 
1.054189 
1.054311 
1.054433 
1.054584 
1.054740 
1.054925 
1.055114 
1.055337 
1.055564 
1.055825 
1.056107 
1.056439 
1.056797 
1.057223 
1.057694 
1.058237 
1.058877 
1.059603 
1.060419 
1.061332 



.007167 
.007193 
.007218 
.007244 
.007269 
.007305 
.007331 
.007357 
.007394 
.007431 
.007469 
.007517 
.007566 
.007615 
.007666 
.007717 
.007779 
.007844 
.007908 
.007985 
.008064 
.008144 
.008236 
.008342 
.008451 
,008561 
.008697 
.008837 
.009003 
.009173 
.009373 
.009577 
.009811 
.010064 
.010362 
.010682 
.011064 
.011484 
.011970 
.012541 
.013188 
.013914 
.014726 



A 



merican 



Life Insurance. 
Table II — Continued. 
Experience Rate of Mortality. J v 



12 7 



100 



104.5 



4>^ PEK CENT. 



> 

II 


o 


66,797 


1,091 


65,706 


1,143 


64,563 


1,199 


63,364 


1,260 


62,104 


1,325 


60,779 


1,394 


59,385 


1,468 


57,917 


1,546 


56,371 


1,628 


54,743 


1,713 


53,030 


1,800 


51,230 


1,889 


49,341 


1,980 


47,361 


2,070 


45,291 


2,158 


43,133 


2,243 


40,890 


2,321 


38,569 


2,391 


36,178 


2,448 


33,730 


2,487 


31,243 


2,505 


28,738 


2,501 


26,237 


2,476 


23,761 


2,431 


21,330 


2,369 


18,961 


2,291 


16,670 


2,196 


14,474 


2,091 


12,383 


1,964 


10,419 


1,816 


8,603 


1,648 


6,955 


1,470 


5,485 


1,292 


4,193 


1,114 


3,079 


933 


2,146 


744 


1,402 


555 


847 


385 


462 


246 


216 


137 


79 


58 


21 


18 


3 


3 



N, 



80386 

73906 

67806 

62071 

56684 

51631 

46900 

42476 

38347 

34501 

30928 

27615 

24552 

21730 

19137 

16765 

14602 

12641 

10870 

9282 

7863 

6606 

5500 

4534 

3696 

2977 

2364 

1850 

1422 

1071 

789 

567 

394 

264 

169 

102 

57 

30 

13 

5 

1 



.97 

.69 

.76 

.043 

.250 

.929 

.322 

.317 

.472 

.891 

.187 

.386 

.846 

.248 

.588 

.011 

.780 

,259 

.751 

.5174 

.6254 

.8334 

.5894 

.1105 

.5299 

.0204 

.9652 

.0347 

.1909 

.9182 

.89158 

.04928 

.65268 

.54838 

.37316 

.49372 

.887413 

.000650 

.878713 

.463610 

,698696 

.381010 

.045822 



D. 



6480 
6099 
5735 
5386 
5052, 
4731, 
4424, 
4128 
3845, 
3573, 
3312, 
3062 , 
2822, 
2592, 
2372, 
2162, 
1961, 
1770. 
1589. 
1417. 
1256, 
1106. 

966. 

837. 

719. 

612. 

514. 

427. 

350. 

282. 

222. 

172. 

130. 

95. 

66. 

44. 

27. 

16. 

8. 

3. 

1. 



276 

936 

716 

793 

321 

607 

004 

846 

581 

704 

801 

540 

598 

660 

577 

231 

521 

508 

234 

892 

792 

244 

4789 

5806 

5095 

0552 

9305 

8438 

2727 

0267 

8423 

3966 

1043 

17522 

87942 

60631 

88676 

12194 

415103 

764914 

317686 

335188 

045822 



12.404850 
12.115991 
11.821859 
11.522829 
11.199405 
10.912104 
10.601303 
10.287712 
9.971829 
9.654386 
9.335966 
9.017152 
8.698695 
8.381449 
8.066151 
7.753570 
7.444624 
7.139897 
6.840261 
6.546001 
6.256916 
5.972338 
5.691364 
5.413345 
5.137566 
4.863860 
4.692799 
4.324116 
4.060268 
3.800730 
3.544611 
3.289214 
3.033356 
2.779593 
2.532516 
2.297740 
2.075802 
1.860858 
1.649262 
1.451184 
1.289151 
1.136706 
1.000000 



1.062351 
1.063499 
1.064763 
1.066202 
1.067780 
1.069530 
1.071486 
1.073659 
1.076077 
1.078756 
1.081716 
1.085007 
1.088686 
1.092761 
1.097282 
1.102322 
1.107884 
1 . 1 14064 
1.120841 
1.128183 
1.136089 
1.144613 
1.153892 
1.164099 
1.175562 
1.188616 
1.203516 
1.221459 
1.241852 
1.265587 
1.292614 
1.325063 
1.366999 
1.423087 
1.499325 
1.599550 
1.729631 
1.915832 
2.235137 
2.857209 
3.931191 
7.244979 



.015629 

.016646 

.017771 

.019029 

.020416 

.021948 

.023655 

.025544 

.027636 

.029944 

.032481 

.035285 

.038400 

.041825 

.045595 

.049762 

.054317 

.059323 

.064751 

.070557 

.076725 

.083280 

.090306 

.097905 

.106281 

.115623 

.126060 

.138245 

.151689 

.166791 

.183312 

.202257 

.225408 

.254240* 

.289971 

.331762 

.378780 

.434972 

.509538 

.606946 

.702562 

.820232 

.956938 



128 



Xotes on 



Table III— Whole Life Policy for %\, 000— Actuaries' Rate of 

100 4 PEK CENT. 



Mortality. \ v = 



104 





1 


o 


1 O 




Age. 


int that will 
1,000 for one 
ifferent ages. 

1000. 


lal Premium 

Insure $1,000 

at different 


Ic Premium 

Insure $1,000 

at different 


d Deposit or 
i " at the end 
Policy Year, 
Die Life Pol- 
n out at dif- 
es. 




1^^ ^ 


Annr 

twill 

Life 

s. 

-(1- 




t^ « * ce - S 




£i£^^ 


ii.^iX 




10 


6.50 


10.429 


213.323 


When insurance is paid 


11 


6.52 


10.632 


216.560 


for year by year, as indi- 


12 


6.55 


10.842 


219.929 


cated in the column of 


13 


6.58 


11.065 


223.438 


premiums headed 


14 


6.63 


11.299 


227.080 


d , , 


15 


6.68 


11.544 


230.860 


ijpXiooo, 

^x 


16 


6.73 


11.800 


234.746 


17 


6.79 


12.067 


238.842 


there is no Deposit on 


18 


6.86 


12.349 


243.040 


hand at the end of any 


19 


6.93 


12.643 


247.403 


year, the amount paid 


20 ' 


7.01 


12.950 


251.908 


each year being just suf- 


21 


7.09 


13.272 


256.565 


ficient to pay cost of in- 


22 


7.18 


13.610 


261.378 


surance during that year. 


23 


7.27 


13.964 


266.358 


When insurance for 


24 


7.37 


14.334 


271.501 


whole life is paid for by a 


25 


7.47 


14.721 


276.817 . 


net single premium, paid 


26 


7.58 


15.129 


282.314 


at the age a;, the Deposit 


27 


7.70 


15.557 


287.992 


at the end of any number 


28 


7.83 


16.005 


293.857 


of years n must be equal 


29 


7.96 


16.476 


299.914 


to the net single premium 


30 


8.10 


16.972 


306.169 


at the age x-\-n. When 


31 


8.25 


17.492 


312.614 


insurance at the age x for 


32 


8.41 


18.039 


319.290 


whole life is paid for by 


33 


8.58 


18.617 


326.168 


equal annual premiums, 


34 


8.75 


19.225 


333.268 


the Deposit that must be 


35 


8.93 


19.866 


340.600 


OD hand at the end of n 


36 


9.12 


20.544 


348.171 


years may be obtained by 


37 


9.31 


21.260 


355.989 


taking the difference be- 


38 


9.52 


22.019 


364.069 


tween the net annual pre- 


39 


9.74 


22.823 


372.415 


mium at the age x-{-n and 


•40 


9.96 


23.677 


381.040 


the net annual premium 


41 


10.20 


24.585 


389.961 


at the age x, and multiply- 


42 


10.48 


25.554 


399.184 


ing this difference by the 


43 


10.82 


26.585 


408.709 


value of large A at the 


44 


11.25 


27.682 


418.516 


age x-\-n^ which, in this 


45 


11.74 


28.846 


428.572 


case, is obtained from 


46 


12.34 


30.080 


438.862 


Table I. 


47 


13.00 


31.385 


449.347 




48 


13.71 


32.766 


460.023 




49 


14.48 


34.228 


470.878 




50 


15.33 


35.775 


481.910 




51 


16.25 


37.415 


493.107 




52 


17.26 


39.153 


504.459 




53 


18.36 


40.996 


515.949 




54 


19.53 


42.949 


527.567 




55 


20.83 


45.025 


539.312 





Life Insurance. 



Table III— Continued.— WM^ Life Policy for $1,000— JLc/tt- 
aries^ Rate of Mortality, j ^ = -J^- * ^^^ ''^''^* 





1 


o 


o 




Age. 


Net Amount that will 
Insure $1,000 for one 
year at different ages. 

t;^XlOOO. 

X 


Net Annual Premium 
thatwilllnsure$l,000 
for Life at different 

ages. 

fi— (I— i^)]Xiooi 


Net Single Premium 
that will Insure$l, 000 
for Life at different 
ages. 

l._(l._^,)AJXlOOi 

\ 1 


Trust Fund Deposit or 
"Reserve" at the end 
of any Policy Year, 
of a Whole Life Pol- 
icy, taken out at dif- 
ferent ages. 


56 


22.24 


47.229 


551.161 


When insurance is paid 


57 


23.73 


49.571 


563.103 


for year by year, as indi- 


58 


25.37 


52.066 


575.142 


cated in the column of 


59 


27.16 


54.723 


587.258 


premiums headed 


60 


29.17 


57.556 


599.430 


// 


61 


31.36 


60.572 


611.628 


j.fXiooo, 

*x 


62 


33.77 


63.778 


623.826 


63 


36.38 


67.198 


635.995 


there is no Deposit oa 


64 


39.25 


70.838 


648.120 


hand at the end of any 


65 


42.39 


74.718 


660.170 


year, the amount paid 


66 


45.78 


78.848 


672.125 


each year being just suf- 


67 


49.49 


83.238 


683.970 


ficient to pay cost of in- 


68 


53.49 


87.918 


695.650 


surance during that year. 


69 


57.78 


92.898 


707.192 


When insurance for 


70 


62.44 


98.198 


718.569 


whole life is paid for by a 


71 


67.46 


103.868 


729.764 


net single premium, paid 


72 


72.89 


109.898 


740.762 


at the age x, the Deposit 


73 


78.73 


116.358 


751.566 


at the end of any number 


74 


85.06 


123.238 


762.152 


of years n must be equal 


75 


91.88 


130.608 


772.514 


to the net single premium 


76 


99.21 


137.488 


782.644 


at the age x-\-n. When 


T7 


107.18 


146.938 


792.546 


insurance at the age x for 


78 


115.81 


155.988 


802.198 


whole life is paid for by 


79 


125.06 


165.678 


811.593 


equal annual premiums, 


80 


135.01 


176.098 


820.742 


the Deposit that must be 


81 


145.61 


187.318 


829.654 


on hand at the end of n 


82 


156.92 


199.478 


838.362 


years may b^ obtained by 


83 


169.15 


212.778 


846.914 


taking the difference be- 


84 


182.38 


227.428 


855.344 


tween the net annual pre- 


85 


197.21 


243.738 


863.707 


mium at the age x-\-ii and 


86 


213.92 


262.104 


872.002 


the net annual premium 


87 


232.92 


282.687 


880.237 


at the age x, and multi- 


88 


255.07 


306.226 


888.417 


plying this difference by 


89 


281.14 


333.138 


896.497 


the value of large A at 


90 


311.28 


363.889 


904.408 


the age x-{-n, which, ia 


91 


347.10 


399.304 


912.142 


tbis case, is obtained from 


92 


389.68 


439.948 


919.606 


Table I. 


93 


439.64 


486.067 


926.675 




94 


496.45 


537.288 


933.198 




95 


501.80 


592.209 


939.064 




96 


623.70 


645.618 


943.777 




97 


665.68 


693.078 


947.424 




98 


721.15 


767.738 


952.293 




99 


961.54 


961.538 


961.538 





ISO 



Jfotes on 



Table IV. 
Whole Life Policy for $1,000 — American Experience Rate of 

Mortality. \v= . 

^ I 104.5 

4>^ PER CENT. 







o 


O 








Sop O 


3 5 « O 


•^ c S o;;5 


Age. 


jnt that 
1,000 for 
itt'erent a 

1000. 




'S -T.CD '"' 
g €^$3 V" 


id Deposi 
3" at the 
Policy Y 
ole Life 
n out at 
;es. 




C <D cj 
1^^ ^ 


Net Sing 
• that will 

for Life 

ages. 

1— (1— i; 


Net Anni 
thatwill 
for Life 
ages. 


Trust Fur 
"Reserv( 
of any 
of a Wh 
icy take 
fereat ag 


10 


7.16 


188.050 


9.972 


When insurance is paid 


11 


7.19 


190.450 


10.130 


for year by year, as indi- 


12 


7.21 


192.953 


10.295 


cated in the column of 


13 


7.24 


195.568 


10.469 


premiums headed 


14 


7.27 


198.300 


10.652 


f ^Xiooo, 


15 


7.30 


201.157 


10.843 


16 


7.33 


204.132 


11.045 


17 


7.36 


207.245 


11.257 


there is no Deposit on 


18 


7.39 


210.501 


11.481 


hand at the end of any 


19 


7.43 


213.899 


11.717 


year, the amount paid 


20 


7.47 


217.448 


11.966 


each year being just suf- 


21 


7.52 


221.155 


12.227 


ficient to pay cost of in- 


22 


7.57 


225.021 


12.503 


surance during that year. 


23 


7.61 


229.050 


12.794 


When insurance for 


24 


7.67 


233.255 


13.100 


whole life is paid for by a 


25 


7.72 


237.643 


13.423 


net single premium, paid 


26 


7.78 


242.226 


13.765 


at the age x, the Deposit 


27 


7.84 


247.003 


14.125 


at the end of any number 


28 


7.91 


251.989 


14.506 


of years n must be equal 


29 


7.98 


257.191 


14.910 


to the net single premium 


30 


8.06 


262.608 


15.336 


at the age x-\-n. When 


31 


8.14 


268.260 


15.787 


insurance at the age x for 


32 


8.24 


274.155 


16.265 


whole life is paid for by 


33 


8.34 


280.299 


16.771 


equal annual premiums. 


34 


8.45 


286.695 


17.308 


the Deposit that must be 


35 


8.56 


293.352 


17.876 


on hand at the end of n 


36 


8.70 


300.295 


18.481 


years may be obtained by 


37 


8.84 


307.514 


19.123 


taking the difference be- 


38 


9.00 


315.429 


19.805 


tween the net annual pre- 


39 


9.17 


322.832 


20.529 


mium at the age x-\-n and 


40 


9.37 


330.948 


21.301 


the net annual premium 


41 


9.58 


339.366 


22.121 


at the age x, and multi- 


42 


9.81 


348.114 


22.996 


plying this difference by 


43 


10.06 


357.193 


23.929 


the value of large A at 


44 


10.36 


366.602 


24.924 


the age x-\-n^ which, in 


45 


10.68 


376.346 


25.986 


this case, is obtained from 


46 


11.06 


386.433 


27.121 


Table II. 


47 


11.48 


396.846 


28.333 




48 


11.97 


407.594 


29.628 




49 


12.54 


418.668 


31.013 




60 


13.18 


430.032 


32.490 




51 


13.91 


441.695 


34.068 




52 


14.73 


453.626 


35.753 





Life Insurance. 



131 







Table IV- 


—Continued. 


Whole Life Policy for $1,000- 


—American Experience Rate of 






Mortaliiy. 


100 

V — 


5 




I 104., 






4% PER CENT. 




will 
one 

ges. 


ium 
,000 
rent 

1000 


a 


,000 
rent 

1000 


t or 
end 
ear, 
Pol- 
dif- 


Age. 


nt that 
L,000 for 
ifferent a 

100. 


S^.^ X 




fi ^ 

a ^ 1 


d Deposi 
;" at the 
Policy Y 
ole Life \ 
Q out at 
es. 




o^2 X 


Sing 

twill 

Life 


C 
< 


t will 
Life 

s. 

-(1- 






^ o3 tj «.> '^— ' 






if-o-M 


53 


15.63 


465.820 




37.551 


When insurance is paid 


54 


16.65 


478.259 




39.473 


for year by year, as indi- 


55 


17.77 


490.925 




41.527 


cated in the column of 


56 


19.03 


503.801 




43.722 


premiums headed 


57 


20.42 


516.866 




46.069 


d 


58 


21.95 


530.099 




48.579 


^^Xiooo, 


59 


23.66 


543.484 




51.265 


60 


25.54 


556.989 




54.141 


there is no Deposit on 


61 


27.64 


570.591 




57.122 


hand at the end of any 


62 


29.94 


584.261 




60.517 


year, the amount paid 


63 


32.48 


597.973 




64.050 


each year being just suf- 


64 


35.28 


611.702 




67.837 


ficient to pay cost of in- 


65 


38.40 


625.415 




71.897 


surance during that year. 


66 


41.82 


639.076 




76.249 


When i nsu ran c e for 


67 


45.59 


652.654 




80.913 


whole life is paid for by 


68 


49.76 


666.114 




85.910 


a net single premium, paid 


69 


54.31 


679.418 




91.263 


at the age z, the Deposit 


70 


59.32 


692.541 




96.996 


at the end of any number 


71 


64.75 


705.443 




103.131 


of years n must be equal 


72 


70.56 


718.115 




109.703 


to the net single premium 


73 


76.72 


730.564 




116.761 


at the :ige x-\-n. When 


74 


83.28 


742.818 




124.377 


insurance at the age x for 


75 


90.31 


754.917 




132.643 


whole life is paid for by 


76 


97.90 


766.889 




141.667 


equal annual premiums, 


77 


106.28 


778.765 




151.583 


the Deposit that must be 


78 


115.62 


790.551 




162.536 


on hand at the end of n 


79 


126.06 


802.224 




174.670 


years may be obtained by 


80 


138.24 


813.794 




188.199 


taking the difference be- 


81 


151.69 


825.156 




203.227 


tween the net annual pre- 


82 


166.79 


836.332 




220.045 


mium at the age x-\-n and 


83 


183.31 


847.361 




239.056 


the net annual premium 


84 


202.26 


858.359 




260.962 


at the age .r, and multi- 


85 


225.41 


869.377 




286.606 


plying this difference by 


86 


254.24 


880.305 




316.703 


the A'alue of large A at 


87 


289.97 


890.944 




351.802 


the age .r+n, which, in 


88 


331.76 


901.543 




392.148 


this case, is obtained from 


89 


378.78 


910.609 




438.668 


Table II. 


90 


434.97 


919.868 




494.324 




91 


509.54 


929.980 




563.270 




92 


606.95 


937.509 




646.031 




93 


702.56 


944.846 




732.643 




94 


820.23 


951.051 




836.673 




95 


956.94 


956.938 




956.938 





INDEX 



PAGE, 

Abbreriations 8 

Accountants . .. 113 

Accounts to be kept "with each policy 81 

Account of a policy for the 10th policy year 85 

Accrued liability . 38 

Accumulation of Deposit or " Reserve " necessary 38, 105 

Actuaries' Table of Mortality 120, 124, 125 

Additional insurance purchased with surplus 87 

Agents' commissions 107 

Age of policy-holders — how considered 111 

Algebraic Summary of the theory of whole life insurance 44 

American experience rate of mortality — Table II 121, 126, 127 

Amount that will produce one dollar in one year 11 

Amount that will insure one dollar for one year 12 

Amount that will produce one dollar in two years 17 

Amount that will produce one dollar in n years 19 

Amount that will insure one dollar for whole life 19 

Amount at Risk during any year 35 

aPx — meaning of this symbol 8 

aPx — formula used for calculating 27, 29, 46 

aFj:\n 53, 57 

Assets— three times the liabilities 99 

Aa; — meaning of this symbol 9 

Aj; — numerical value of 27, 50, 118 

Axln — meaning and value 51 

Barrett, George 119 

Borrowing great names 97 

Calculation of Cost of Insurance 37, 47 

Calculation of Surplus 83 

Certainty of payment of policy at maturity desirable 80 

Comments on Amount in Deposit, or Reserve 41 

Comments on Insurance partly upon credit 95 

Companies the custodians of immense suras of money 107 

Companies not so rich as some would suppose 110 

Comparison between a Cash Company and a Xote Company that retains 20 

per cent, of the surplus 90 

Contribution plan 77 

Cost of Insurance 34, 78 

Cost of Insurance and Annual Premium compared 37 

Davies, Griffith 120 



Index. 133 

/ 

PAGE. 

Deposit or Reserve 29 

Deposit or Reserve absolutely necessary 37 

Deposit or Reserve — example for calculating deposits 84 

Deposit or Reserve not cash capital 99, 108 

Deposit by companies of $100,000 not a certain safeguard 106 

Difference between the general law upon which the risk in Life Insurance is 

based, and the nature of the risk in Fire Insurance 112 

Dividends cannot arise from net annual premiums at net rate of interest 43 

Dividends to Policy-holders being surplus arising from over-payment, 81, 85, 93^ 

98 

Dividends to Shareholders sometimes enormous 102, 103 

Dx column explained 28, 50, 118 

Effect of grouping a policy for a large amount with policies of smaller 

amount "79 

Endowment 54 

Endowment — Formulas for calculating 116 

Endowment and Term Insurance combined 55 

Endowment — Example of calculating 55 

Endowment and Term Insurance — example of calculating 56 

Endowment and Term Insurance — Formulas for calculating net single and 

net annual premiums 117 

Equation of Equitable balance 36 

Ex (ft — meaning of this symbol 54 

E^|n — Formula for calculating 54 

Expenses — Loading — Surplus 75, 79 

Explanation of manner in which Tables herein were constructed 118 

Fire Insurance compared with Life Insurance 112 

Formula for computing the net single premium 26 

Formula for computing the net annual premium 27 

Formula for computing the premiums in Endowment 56 

Formula for computing the Deposit or Reserve in whole Life Insurance— 33, 34 

Formula (general) for computing the Deposit or Reserve . 58, 60, 63 

Fraction that represents the chance or probability that the Insured may die 

any year 13 

Fraction that represents the cjjiance or probability that the Insured may be 

alive at the end of any year 23 

Further allusion to Loading and Expenses 79 

General comments 97 

General Summary of Formulas and Arithmetical rules for Life Insurance net 

calculations 115 

Grouping policies 79 

"Happy Capitalists" 74, 102, 103 

Haste in advertising payment of policy 1 n 

Impaired or diseased lives 113 

Insurance of $1 for one year at age 30 15 

Insurance for each separate year 21 

Insurance of $10,000 for one year, age 40 — 78,653 persons insured, and the 

result 16 



1SI{^ Index. 



PAGE. 

Insurance by net single premium for whole life 19 

Insurance by net annual premium for whole life 8, 22 

Insurance other than whole life 50 

Insurance partly on credit 95, 104 

Insuring for each separate year, and insuring by net single premium for 

whole life in advance . 21 

Introduction 4 

Large deposit or "reserve" means large debt 108 

Lawyers should understand principles of Life Insurance 113 

Liabilities 99 

Life Insurance may be made safe . 106 

Life Insurance Companies great money lenders 110 

Life Insurance compared with Fire Insurance 112 

Life Assurance, quotation from Gladstone concerning 74 

Life series of annual premiums of $1 each; value 8, 23 

Loading 75, 79 

Management of Life Insurance Companies 109 

Medical Examiners 112 

Mixed Companies 76 

Mutual Companies „ 75, 103-4 

Net annual premium, whole life insurance 22 

Net annual premium that will insure $1 for life, at age 42 29 

Net annual premium — Formula to insure $1 for whole life 115 

Net annual premium — Term Insurance — Examples 53 

Net annual premium that will insure §1 for n years — Formula 116 

Net annual premium for n years that will insure $1 for whole life 117 

Net annual premium — Term Insurance and Endowment — Examples 1 55 

Net single premium — Whole Life Insurance 8, 19 

Net single premium — Formula to insure $1 for whole life 115 

Net single premium — Term Insurance — Examples 53 

Net single premium — To insure $1 for n years 116 

Net single and net annual premium — relation between.: 25 

n annual payments of $1 each 50 

No dividends from net premiums at net interest 100 

Not exempt from usual result of mismanagement in business 109 

Notes on Life Insurance , 7 

Numerical bragging . 98 

Nx column „ 28, 50, 118 

One hundred thousand dollars deposit is of but little avail in certain cases 106 

Part second 75 

Policy-holders should investigate certain points 100 

Policy-holders assumed to be aged an exact number of whole years 111 

Practical Life Insurance 75 

Practice in payment of Policies 111 

Price charged 101, 102 

Principles upon which money values in Life Insurance are calculated impor- 
tant to be understood 109 

"Problem that stands at the threshold of Life Insurance" — and something 
within the threshold 69 



Index. 135 

PAGE, 

Professors of High Schools — 113 

Profits 103 

Proprietary or purely stock compi'vDies 105 

Quotations — Dr. Farr Title page 

Prof. Elizur Wright 3, 74 

Prof. D. Parks Fackler 3 

Hon. John E. Sandford 3 

Chancellor Gladstone 74 

Actuary of the Royal Exchange 74 

Morgan . 114 

William Barnes 114 

Higham 114 

Dr. Price 114 

r — different meanings of this symbol 11, 59, 120 

Rate of mortality — variation in 78 

Ratio — meaning of the term 59 

Registered policies 42 

Relation existing between sV^ and A^, 25 

Reserve called in these Notes Deposit 29 

Reserve absolutely necessary 37 

Return premium plan 96 

s?x — meaning of this symbol 8 

sVx — Formula used for calculating 26, 45, 46 

«Px U — meaning of this and formula for calculating 52 

(5P-1-E)xln — meaning of this and formula for calculating 56 

Stock Companies 105 

Surplus 75, 81, 83 

Tables I, II, III, IV 120, 121 

Table I 124, 125 

Table II 126, 127 

Table III 128, 129 

Table IV 130, 131 

Term Insurance 52 

Term Insurance — Example of calculating 53, 116 

Tetens, John Nicholas . 119 

T. F. D. — meaning of this symbol 9 

Trust Fund Deposit or Reserve— (T. P. D.)a;+n— 29, 38, 40, 58, 63, 115, 121 
(T. F. D.)a;+«— Table III and Table IV, for calculating, 121, 128, 129, 130, 131 

Theory of Life Insurance 10 

V — meaning of this symbol 11 

V — method of calculating 11, 120 

^// y/// ^//// ^(._ — meaning of these symbols 17 

i// ^/z/ y//// &c.— method of calculating 17, 18, 19 

Valuation tables described GO, 121 

Value of a life series of annual premiums or payments of one dollar each 23 

Value of a life series of annual premiums or payments of $1 each, the first 
payment to be made at the end of n years 51 



136 Index. 

PAGB. 

Value of a policy at the end of a policy year is sometimes greater than the 

value at the beginning of same policy year 67 

Value of a policy for fractional parts of a year — method of calculating 65 

Variations from the table rate of mortality 78 

Whole Life Insurance — net single premium 19 

Whole Life Insurance — net annual premium 21 

Whole Life Insurance — Deposit or "Reserve" 29 

Whole Life Insurance — Cost of Insurance 34 

Whole Life Insurance — Algebraic summary 44 

Whole Life Insurance — paid for in n years 57 



SECOND LESSON IN LIFE INSURANCE CALCULATIONS. 

To determine the amount of money that will, at a given 
rate of interest, produce $1 in any given number of years, 
the interest being compounded annually: raise the expression 
that gives the amount that will produce $1 in one year, at the 
same rate of interest, to a power indicated by the number of 
years. If the rate of interest is 4 per cent., the amount of 
money that will produce $1, in one year, is 100 divided by 104; 
which is equal to the decimal $0.96153846153846 (and the 
series of decimal figures 153846 will be repeated to infinity). 
If we multiply 10.96153846 by 0.96153846, we obtain the 
amount that will, if invested at 4 per cent, compound interest, 
produce $1 in two years. Multiply this result by 0.96153846, 
and we obtain the amount that will, if invested for three years 
at 4 per cent, compound interest, produce $1 at the end of 
that time ; and so on for other years. 

Suppose that the rate of interest is 4J per cent.; that the 
person to be insured is 20 years old ; and that he desires to 
have $1 insured, to be paid to his heirs at the end of 30 years, 
provided he dies between age 49 and age 50. How much 
money in hand, at age 20, will be required to pay for the pro- 
posed insurance? Divide 100 by 104^; then raise this quan- 
tity to the 30th power. This gives the amount that will, at 4^ 
per cent, compound interest, produce $1 in 30 years. From 
the Mortality Table obtain the number of deaths between age 
49 and age 50 : it is 927. Divide this by the number living at 

927 
age 20; this number living is 92,637. Then is the 

J/4,uo7 

fraction that represents, at age 20, the chance or probability 
at that time that the insured will die between the age 49 and 
age 50. Multiply the amount that will produce $1 in 30 years 
by the fraction which represents the chance that the ins^ured 
will die during the 30th year from that time, and we obtain 
the amount that will insure $1, to be paid to the heirs of the 
insured at the end of 30 years, in case he dies between 49 
and 50 years of age. In like manner the calculation is made 
at- any age, for insurance during any named year. At any 
age, if the calculation is made for insurance during every 
year from that age to the limit of the Table, the sum of all 
these respective yearly amounts will, at that age, effect the 
insurance for whole life. 



FIRST LESSON IN LIFE INSURANCE CALCULATIONS. 

To obtain at any age the amount that will insure $1,000 to be paid to the heirs 
of the insured at the end of one year, in case the insured dies during the year: 
a Table showing the Rate of Mortality must be furnished, and a rate of interest 
fixed upon. Assume that the Table is that which purports to give the rate of 
mortality among insured lives in this country, which is called 

American Experience Rate of Mortality. 



> 

a> to 




32 


100,000 


749 


99,251 


746 


33 


98,505 


743 


34 


97,762 


740 


35 


97,022 


737 


36 


96,285 


735 


37 


95,550 


732 


38 


94,818 


729 


39 


94,089 


727 


40 


93,362 


725 


41 


92,637 


723 


42 


91,914 


722 


43 


91,192 


721 


44 


90,471 


720 


45 


89,751 


719 


46 


89,032 


718 


47 


88,344 


718 


48 


87,596 


718 


49 


86,878 


718 


50 


86,160 


719 


51 


85,441 


720 


52 


84,721 


721 


53 





o 


< 


84,000 


723 


54 


83,277 


726 


55 


82,551 


729 


56 


81,822 


732 


57 


81,090 


737 


58 


80,353 


742 


59 


79,611 


749 


60 


78,862 


756 


61 


78,106 


765 


62 


77,341 


774 


63 


76,567 


785 


64 


75,782 


797 


65 


74,985 


812 


66 


74,173 


828 


67 


73,345 


848 


68 


72,497 


870 


69 


71,627 


896 


70 


70,731 


927 


71 


69,804 


962 


72 


68,842 


1,001 


73 


67,841 


1,044 


74 


66,797 


1,091 


75 



p — 

5 

;2; 



65,706 
64,563 
63,364 
62,104 
60,779 
59,385 
57,917 
56,371 
54,743 
53,030 
51,230 
49,341 
47,361 
45,2.91 
43,133 
40,890 
38,569 
36,178 
33,730 
31,243 
28,738 
26,237 



^ 




-5 


< 


1,143 


76 


1,199 


77 


1,260 


78 


1,325 


79 


1,394 


80 


1,468 


81 


1,546 


82 


1,628 


83 


1,713 


84! 


1,800 


85, 


1,889 


86 


1,980 


87 


2,070 


88 


2,158 


89 


2 243 


90 


2,321 


91 


2,391 


92 


2,448 


93 


2,487 


94 


2,505 


95 


2,501 




2,476 





a 



23,761 

21,330 

18,961 

16,670 

14,474 

12,383 

10,419 

8,603 

6,955 

5,485 

4,193 

3,079 

2,146 

1,402 

847 

462 

216 

79 

21 

3 



2,431 

2,369 

2,291 

2,196 

2,091 

1,964 

1,816 

1,648 

1,470 

1,292 

1,114 

933 

744 

555 

385 

246 

137 

58 

18 

3 



• — ] n ; 

After having assumed a rate of interest, we obtain the amount that will, at this 
rate, produce $1 in one year, bj dividing 100 by 100, plus the rate of interest. 
Suppose the interest is assumed to be seven per cent., and that the person to be 
insured for one year is aged 50. The amount that will, if paid in advance, and 
invested at 7 per cent., produce $1 certain in one year, when principal and inter- 
est at this rate for one year are added together, is obtained by dividing 100 by 
107. This makes $0.934579. Then multiply this amount by the number of deaths 
given in the Table opposite to age 50, which is 962, and divide the product by the 
number living at the same age, which is 69,804. The result is $0.012879. This 
is the amount that will insure $1 for one year, if paid in hand at age 50. Cyie 
thousand times this amount; or $12.88, will insure $1,000 for one year at the same 
age. In a precisely similar manner, the calculations are made for insurance for 
one year at any age, and for any amount, and at any rate of interest. 

See inside page of cover. 



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